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Journal articles on the topic 'Monoid ring'

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Published: 4 June 2021

Last updated: 2 February 2022

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1

COJUHARI, E. P., and B. J. GARDNER. "GENERALIZED HIGHER DERIVATIONS." Bulletin of the Australian Mathematical Society 86, no. 2 (January 6, 2012): 266–81. http://dx.doi.org/10.1017/s000497271100308x.

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AbstractA type of generalized higher derivation consisting of a collection of self-mappings of a ring associated with a monoid, and here called a D-structure, is studied. Such structures were previously used to define various kinds of ‘skew’ or ‘twisted’ monoid rings. We show how certain gradings by monoids define D-structures. The monoid ring defined by such a structure corresponding to a group-grading is the variant of the group ring introduced by Năstăsescu, while in the case of a cyclic group of order two, the form of the D-structure itself yields some gradability criteria of Bakhturin and Parmenter. A partial description is obtained of the D-structures associated with infinite cyclic monoids.

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2

Alhevaz, Abdollah, Ebrahim Hashemi, and Michał Ziembowski. "Nilradicals of the unique product monoid rings." Journal of Algebra and Its Applications 16, no. 07 (July 7, 2016): 1750133. http://dx.doi.org/10.1142/s021949881750133x.

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Armendariz rings are generalization of reduced rings, and therefore, the set of nilpotent elements plays an important role in this class of rings. There are many examples of rings with nonzero nilpotent elements which are Armendariz. Observing structure of the set of all nilpotent elements in the class of Armendariz rings, Antoine introduced the notion of nil-Armendariz rings as a generalization, which are connected to the famous question of Amitsur of whether or not a polynomial ring over a nil coefficient ring is nil. Given an associative ring [Formula: see text] and a monoid [Formula: see text], we introduce and study a class of Armendariz-like rings defined by using the properties of upper and lower nilradicals of the monoid ring [Formula: see text]. The logical relationship between these and other significant classes of Armendariz-like rings are explicated with several examples. These new classes of rings provide the appropriate setting for obtaining results on radicals of the monoid rings of unique product monoids and also can be used to construct new classes of nil-Armendariz rings. We also classify, which of the standard nilpotence properties on polynomial rings pass to monoid rings. As a consequence, we extend and unify several known results.

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3

HABIBI, MOHAMMAD, and RAOUFEH MANAVIYAT. "A GENERALIZATION OF NIL-ARMENDARIZ RINGS." Journal of Algebra and Its Applications 12, no. 06 (May 9, 2013): 1350001. http://dx.doi.org/10.1142/s0219498813500011.

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Let R be a ring, M a monoid and ω : M → End (R) a monoid homomorphism. The skew monoid ring R * M is a common generalization of polynomial rings, skew polynomial rings, (skew) Laurent polynomial rings and monoid rings. In the current work, we study the nil skew M-Armendariz condition on R, a generalization of the standard nil-Armendariz condition from polynomials to skew monoid rings. We resolve the structure of nil skew M-Armendariz rings and obtain various necessary or sufficient conditions for a ring to be nil skew M-Armendariz, unifying and generalizing a number of known nil Armendariz-like conditions in the aforementioned special cases. We consider central idempotents which are invariant under a monoid endomorphism of nil skew M-Armendariz rings and classify how the nil skew M-Armendariz rings behaves under various ring extensions. We also provide rich classes of skew monoid rings which satisfy in a condition nil (R * M) = nil (R) * M. Moreover, we study on the relationship between the zip and weak zip properties of a ring R and those of the skew monoid ring R * M.

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4

Paykan, Kamal, and Ahmad Moussavi. "The McCoy condition on skew monoid rings." Asian-European Journal of Mathematics 10, no. 03 (September 2017): 1750050. http://dx.doi.org/10.1142/s1793557117500504.

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Let [Formula: see text] be an associative ring with identity, [Formula: see text] a monoid and [Formula: see text] a monoid homomorphism. When [Formula: see text] is a u.p.-monoid and [Formula: see text] is a reversible [Formula: see text]-compatible ring, then we observe that [Formula: see text] satisfies a McCoy-type property, in the context of skew monoid ring [Formula: see text]. We introduce and study the [Formula: see text]-McCoy condition on [Formula: see text], a generalization of the standard McCoy condition from polynomial rings to skew monoid rings. Several examples of reversible [Formula: see text]-compatible rings and also various examples of [Formula: see text]-McCoy rings are provided. As an application of [Formula: see text]-McCoy rings, we investigate the interplay between the ring-theoretical properties of a general skew monoid ring [Formula: see text] and the graph-theoretical properties of its zero-divisor graph [Formula: see text].

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5

HABIBI, M., and A. MOUSSAVI. "NILPOTENT ELEMENTS AND NIL-ARMENDARIZ PROPERTY OF MONOID RINGS." Journal of Algebra and Its Applications 11, no. 04 (July 31, 2012): 1250080. http://dx.doi.org/10.1142/s0219498812500806.

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Antoine [Nilpotent elements and Armendariz rings, J. Algebra 319(8) (2008) 3128–3140] studied the structure of the set of nilpotent elements in Armendariz rings and introduced nil-Armendariz rings. For a monoid M, we introduce nil-Armendariz rings relative to M, which is a generalization of nil-Armendariz rings and we investigate their properties. This condition is strongly connected to the question of whether or not a monoid ring R[M] over a nil ring R is nil, which is related to a question of Amitsur [Algebras over infinite fields, Proc. Amer. Math. Soc.7 (1956) 35–48]. This is true for any 2-primal ring R and u.p.-monoid M. If the set of nilpotent elements of a ring R forms an ideal, then R is nil-Armendariz relative to any u.p.-monoid M. Also, for any monoid M with an element of infinite order, M-Armendariz rings are nil M-Armendariz. When R is a 2-primal ring, then R[x] and R[x, x-1] are nil-Armendariz relative to any u.p.-monoid M, and we have nil (R[M]) = nil (R)[M].

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6

Dumitru, Mariana, Laura Năstăsescu, and Bogdan Toader. "Graded near-rings." Analele Universitatii "Ovidius" Constanta - Seria Matematica 24, no. 1 (January 1, 2016): 201–16. http://dx.doi.org/10.1515/auom-2016-0011.

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AbstractIn this paper, we consider graded near-rings over a monoid G as generalizations of graded rings over groups, and study some of their basic properties. We give some examples of graded near-rings having various interesting properties, and we define and study the Gop-graded ring associated to a G-graded abelian near-ring, where G is a left cancellative monoid and Gop is its opposite monoid. We also compute the graded ring associated to the graded near-ring of polynomials (over a commutative ring R) whose constant term is zero.

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7

Singh, Amit Bhooshan. "TRIANGULAR MATRIX REPRESENTATION OF SKEW GENERALIZED POWER SERIES RINGS." Asian-European Journal of Mathematics 05, no. 04 (December 2012): 1250027. http://dx.doi.org/10.1142/s1793557112500271.

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Let R be a ring, (S, ≤) a strictly ordered monoid and ω : S → End (R) a monoid homomorphism. In this paper, we study the triangular matrix representation of skew generalized power series ring R[[S, ω]] which is a compact generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomials rings, (skew) Laurent power series rings, (skew) group rings, (skew) monoid rings, Mal'cev–Neumann rings and generalized power series rings. We investigate that if R is S-compatible and (S, ω)-Armendariz, then the skew generalized power series ring has same triangulating dimension as R. Furthermore, if R is a PWP ring, then skew generalized power series is also PWP ring.

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8

ZHAO, RENYU. "LEFT APP-RINGS OF SKEW GENERALIZED POWER SERIES." Journal of Algebra and Its Applications 10, no. 05 (October 2011): 891–900. http://dx.doi.org/10.1142/s0219498811005014.

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A ring R is called a left APP-ring if the left annihilator lR(Ra) is right s-unital as an ideal of R for any a ∈ R. Let R be a ring, (S, ≤) be a commutative strictly ordered monoid and ω: S → End (R) be a monoid homomorphism. The skew generalized power series ring [[RS, ≤, ω]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings and Malcev–Neumann Laurent series rings. We study the left APP-property of the skew generalized power series ring [[RS, ≤, ω]]. It is shown that if (S, ≤) is a commutative strictly totally ordered monoid, ω: S→ Aut (R) a monoid homomorphism and R a ring satisfying the descending chain condition on right annihilators, then [[RS, ≤, ω]] is left APP if and only if for any S-indexed subset A of R, the ideal lR(∑a ∈ A ∑s ∈ S Rωs (a)) is right s-unital.

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9

Mazurek, Ryszard. "Rota–Baxter operators on skew generalized power series rings." Journal of Algebra and Its Applications 13, no. 07 (May 2, 2014): 1450048. http://dx.doi.org/10.1142/s0219498814500480.

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Let R be a ring, S a strictly ordered monoid, and ω : S → End (R) a monoid homomorphism. The skew generalized power series ring R[[S, ω]] is a common generalization of (skew) polynomial rings, (skew) Laurent polynomial rings, (skew) power series rings, (skew) Laurent series rings, (skew) monoid rings, (skew) Mal'cev–Neumann series rings, and generalized power series rings. We characterize those subsets T of S for which the cut-off operator with respect to T is a Rota–Baxter operator on the ring R[[S, ω]]. The obtained results provide a large class of noncommutative Rota–Baxter algebras.

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10

Sharma, R. K., and Amit B. Singh. "Unified Extensions of Strongly Reversible Rings and Links with Other Classic Ring Theoretic Properties." Journal of the Indian Mathematical Society 85, no. 3-4 (June 1, 2018): 434. http://dx.doi.org/10.18311/jims/2018/20986.

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Let R be a ring, (M, ≤) a strictly ordered monoid and ω : M → <em>End</em>(R) a monoid homomorphism. The skew generalized power series ring R[[M; ω]] is a compact generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomials rings, (skew) Laurent power series rings, (skew) group rings, (skew) monoid rings, Mal'cev Neumann rings and generalized power series rings. In this paper, we introduce concept of strongly (M, ω)-reversible ring (strongly reversible ring related to skew generalized power series ring R[[M, ω]]) which is a uni ed generalization of strongly reversible ring and study basic properties of strongly (M; ω)-reversible. The Nagata extension of strongly reversible is proved to be strongly reversible if R is Armendariz. Finally, it is proved that strongly reversible ring strictly lies between reduced and reversible ring in the expanded diagram given by Diesl et. al. [7].

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11

Okon, James S., and J. Paul Vicknair. "One-Dimensional Monoid Rings with n-Generated Ideals." Canadian Mathematical Bulletin 36, no. 3 (September 1, 1993): 344–50. http://dx.doi.org/10.4153/cmb-1993-047-3.

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AbstractA commutative ring R is said to have the n-generator property if each ideal of R can be generated by n elements. Rings with the n-generator property have Krull dimension at most one. In this paper we consider the problem of determining when a one-dimensional monoid ring R[S] has the n-generator property where R is an artinian ring and S is a commutative cancellative monoid. As an application, we explicitly determine when such monoid rings have the three-generator property.

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12

Gould, Victoria. "Coherent Monoids." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 53, no. 2 (October 1992): 166–82. http://dx.doi.org/10.1017/s1446788700035771.

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AbstractThis paper is concerned with a new notion of coherency for monoids. A monoid S is right coherent if the first order theory of right S-sets is coherent; this is equivalent to the property that every finitely generated S-subset of every finitely presented right S-set is finitely presented. If every finitely generated right ideal of S is finitely presented we say that S is weakly right coherent. As for the corresponding situation for modules over a ring, we show that our notion of coherency is related to products of flat left S-sets, although there are some marked differences in behaviour from the case for rings. Further, we relate our work to ultraproducts of flat left S-sets and so to the question of axiomatisability of certain classes of left S-sets.We show that a monoid S is weakly right coherent if and only if the right annihilator congruence of every element is finitely generated and the intersection of any two finitely generated right ideals is finitely generated. A similar result describes right coherent monoids. We use these descriptions to recognise several classes of (weakly) right coherent monoids. In particular we show that any free monoid is weakly right (and left) coherent and any free commutative monoid is right (and left) coherent.

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13

Mazurek, Ryszard. "Left principally quasi-Baer and left APP-rings of skew generalized power series." Journal of Algebra and Its Applications 14, no. 03 (November 7, 2014): 1550038. http://dx.doi.org/10.1142/s0219498815500383.

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Let R be a ring, S a strictly ordered monoid, and ω : S → End (R) a monoid homomorphism. The skew generalized power series ring R[[S, ω]] is a common generalization of (skew) polynomial rings, (skew) Laurent polynomial rings, (skew) power series rings, (skew) Laurent series rings, and (skew) monoid rings. We characterize when a skew generalized power series ring R[[S, ω]] is left principally quasi-Baer and under various finiteness conditions on R we characterize when the ring R[[S, ω]] is left APP. As immediate corollaries we obtain characterizations for all aforementioned classical ring constructions to be left principally quasi-Baer or left APP. Such a general approach not only gives new results for several constructions simultaneously, but also serves the unification of already known results.

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14

Smertnig, Daniel. "Factorizations in Bounded Hereditary Noetherian Prime Rings." Proceedings of the Edinburgh Mathematical Society 62, no. 2 (November 19, 2018): 395–442. http://dx.doi.org/10.1017/s0013091518000305.

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AbstractIf H is a monoid and a = u1 ··· uk ∈ H with atoms (irreducible elements) u1, … , uk, then k is a length of a, the set of lengths of a is denoted by Ⅼ(a), and ℒ(H) = {Ⅼ(a) | a ∈ H} is the system of sets of lengths of H. Let R be a hereditary Noetherian prime (HNP) ring. Then every element of the monoid of non-zero-divisors R• can be written as a product of atoms. We show that if R is bounded and every stably free right R-ideal is free, then there exists a transfer homomorphism from R• to the monoid B of zero-sum sequences over a subset Gmax(R) of the ideal class group G(R). This implies that the systems of sets of lengths, together with further arithmetical invariants, of the monoids R• and B coincide. It is well known that commutative Dedekind domains allow transfer homomorphisms to monoids of zero-sum sequences, and the arithmetic of the latter has been the object of much research. Our approach is based on the structure theory of finitely generated projective modules over HNP rings, as established in the recent monograph by Levy and Robson. We complement our results by giving an example of a non-bounded HNP ring in which every stably free right R-ideal is free but which does not allow a transfer homomorphism to a monoid of zero-sum sequences over any subset of its ideal class group.

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15

Mazurek, Ryszard. "Archimedean domains of skew generalized power series." Forum Mathematicum 32, no. 4 (July 1, 2020): 1075–93. http://dx.doi.org/10.1515/forum-2019-0187.

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AbstractA skew generalized power series ring {R[[S,\omega,\leq]]} consists of all functions from a strictly ordered monoid {(S,\leq)} to a ring R whose support is artinian and narrow, with pointwise addition, and with multiplication given by convolution twisted by an action ω of the monoid S on the ring R. Special cases of this ring construction are skew polynomial rings, skew Laurent polynomial rings, skew power series rings, skew Laurent series rings, skew monoid rings, skew group rings, skew Mal’cev–Neumann series rings, the “unskewed” versions of all of these, and generalized power series rings. In this paper, we characterize the skew generalized power series rings {R[[S,\omega,\leq]]} that are left (right) Archimedean domains in the case where the order {\leq} is total, or {\leq} is semisubtotal and the monoid S is commutative torsion-free cancellative, or {\leq} is trivial and S is totally orderable. We also answer four open questions posed by Moussavi, Padashnik and Paykan regarding the rings in the title.

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16

Mazurek, Ryszard, and Michał Ziembowski. "On semilocal, Bézout and distributive generalized power series rings." International Journal of Algebra and Computation 25, no. 05 (August 2015): 725–44. http://dx.doi.org/10.1142/s0218196715500174.

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Let R be a ring, and let S be a strictly ordered monoid. The generalized power series ring R[[S]] is a common generalization of polynomial rings, Laurent polynomial rings, power series rings, Laurent series rings, Mal'cev–Neumann series rings, monoid rings and group rings. In this paper, we examine which conditions on R and S are necessary and which are sufficient for the generalized power series ring R[[S]] to be semilocal right Bézout or semilocal right distributive. In the case where S is a strictly totally ordered monoid we characterize generalized power series rings R[[S]] that are semilocal right distributive or semilocal right Bézout (the latter under the additional assumption that S is not a group).

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17

ZHAO, LIANG, and YIQIANG ZHOU. "GENERALISED ARMENDARIZ PROPERTIES OF CROSSED PRODUCT TYPE." Glasgow Mathematical Journal 58, no. 2 (July 21, 2015): 313–23. http://dx.doi.org/10.1017/s001708951500021x.

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AbstractLet R be a ring and M a monoid with twisting f:M × M → U(R) and action ω: M→ Aut(R). We introduce and study the concepts of CM-Armendariz and CM-quasi-Armendariz rings to generalise various Armendariz and quasi-Armendariz properties of rings by working on the context of the crossed product R*M over R. The following results are proved: (1) If M is a u.p.-monoid, then any M-rigid ring R is CM-Armendariz; (2) if I is a reduced ideal of an M-compatible ring R with M a strictly totally ordered monoid, then R/I being CM-Armendariz implies that R is CM-Armendariz; (3) if M is a u.p.-monoid and R is a semiprime ring, then R is CM-quasi-Armendariz. These results generalise and unify many known results on this subject.

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18

GOULD, VICTORIA, and MIKLÓS HARTMANN. "Coherency, free inverse monoids and related free algebras." Mathematical Proceedings of the Cambridge Philosophical Society 163, no. 1 (September 9, 2016): 23–45. http://dx.doi.org/10.1017/s0305004116000505.

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AbstractA monoid S is right coherent if every finitely generated subact of every finitely presented right S-act is finitely presented. This is the non-additive notion corresponding to that for a ring R stating that every finitely generated submodule of every finitely presented right R-module is finitely presented. For monoids (and rings) right coherency is an important finitary property which determines, amongst other things, the existence of a model companion of the class of right S-acts (right R-modules) and hence that the class of existentially closed right S-acts (right R-modules) is axiomatisable.Choo, Lam and Luft have shown that free rings are right (and left) coherent; the authors, together with Ruškuc, have shown that (free) groups, free commutative monoids and free monoids have the same properties. It is then natural to ask whether other free algebras in varieties of monoids, possibly with an augmented signature, are right coherent. We demonstrate that free inverse monoids are not.Munn described the free inverse monoid FIM(Ω) on Ω as consisting of birooted finite connected subgraphs of the Cayley graph of the free group on Ω. Sitting within FIM(Ω) we have free algebras in other varieties and quasi-varieties, in particular the free left ample monoid FLA(Ω) and the free ample monoid FAM(Ω). The former is the free algebra in the variety of unary monoids corresponding to partial maps with distinguished domain; the latter is the two-sided dual. For example, FLA(Ω) is obtained from FIM(Ω) by considering only subgraphs with vertices labelled by elements of the free monoid on Ω.The main objective of the paper is to show that FLA(Ω) is right coherent. Furthermore, by making use of the same techniques we show that FIM(Ω), FLA(Ω) and FAM(Ω) satisfy (R), (r), (L) and (l), related conditions arising from the axiomatisability of certain classes of right S-acts and of left S-acts.

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19

Alonso, Juan M., and Susan M. Hermiller. "Homological Finite Derivation Type." International Journal of Algebra and Computation 13, no. 03 (June 2003): 341–59. http://dx.doi.org/10.1142/s0218196703001407.

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In 1987, Squier defined the notion of finite derivation type for a finitely presented monoid. To do this, he associated a 2-complex to the presentation. The monoid then has finite derivation type if, modulo the action of the free monoid ring, the 1-dimensional homotopy of this complex is finitely generated. Cremanns and Otto showed that finite derivation type implies the homological finiteness condition left FP3, and when the monoid is a group, these two properties are equivalent. In this paper we define a new version of finite derivation type, based on homological information, together with an extension of this finite derivation type to higher dimensions, and show connections to homological type FPnfor both monoids and groups.

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20

Peng, Zhaiming, Qinqin Gu, and Liang Zhao. "Extensions of strongly reflexive rings." Asian-European Journal of Mathematics 08, no. 04 (November 17, 2015): 1550078. http://dx.doi.org/10.1142/s1793557115500783.

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This is a further study of reflexive rings over polynomial rings and monoid rings. The concepts of strongly reflexive rings and strongly [Formula: see text]-reflexive rings are introduced and investigated. Some characterizations of various extensions of the two classes of rings are obtained. It is proved that a ring [Formula: see text] is strongly reflexive if and only if [Formula: see text] is strongly reflexive if and only if [Formula: see text] is strongly reflexive. For a right Ore ring [Formula: see text] with classical right quotient ring [Formula: see text], we show that [Formula: see text] is strongly reflexive if and only if [Formula: see text] is strongly reflexive. Moreover, we prove that if [Formula: see text] is a unique product monoid (u.p.-monoid) and [Formula: see text] is a reduced ring, then [Formula: see text] is strongly [Formula: see text]-reflexive. It is shown that finite direct sums of strongly [Formula: see text]-reflexive rings are strongly [Formula: see text]-reflexive.

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21

FACCHINI, ALBERTO, and FRANZ HALTER-KOCH. "PROJECTIVE MODULES AND DIVISOR HOMOMORPHISMS." Journal of Algebra and Its Applications 02, no. 04 (December 2003): 435–49. http://dx.doi.org/10.1142/s0219498803000593.

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We study some applications of the theory of commutative monoids to the monoid [Formula: see text] of all isomorphism classes of finitely generated projective right modules over a (not necessarily commutative) ring R.

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22

GUO, LI, and ZHONGKUI LIU. "ROTA–BAXTER OPERATORS ON GENERALIZED POWER SERIES RINGS." Journal of Algebra and Its Applications 08, no. 04 (August 2009): 557–64. http://dx.doi.org/10.1142/s0219498809003515.

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An important instance of Rota–Baxter algebras from their quantum field theory application is the ring of Laurent series with a suitable projection. We view the ring of Laurent series as a special case of generalized power series rings with exponents in an ordered monoid. We study when a generalized power series ring has a Rota–Baxter operator and how this is related to the ordered monoid.

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23

Alhevaz, A., and A. Moussavi. "On Monoid Rings Over Nil Armendariz Ring." Communications in Algebra 42, no. 1 (October 18, 2013): 1–21. http://dx.doi.org/10.1080/00927872.2012.657382.

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24

Cojuhari, E. P., and B. J. Gardner. "Skew ring extensions and generalized monoid rings." Acta Mathematica Hungarica 154, no. 2 (January 22, 2018): 343–61. http://dx.doi.org/10.1007/s10474-018-0787-x.

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25

Habibi, Mohammad. "A new class of non-semiprime quasi-Armendariz rings." Studia Scientiarum Mathematicarum Hungarica 51, no. 2 (June 1, 2014): 165–71. http://dx.doi.org/10.1556/sscmath.51.2014.2.1273.

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Hirano [On annihilator ideals of a polynomial ring over a noncommutative ring, J. Pure Appl. Algebra, 168 (2002), 45–52] studied relations between the set of annihilators in a ring R and the set of annihilators in a polynomial extension R[x] and introduced quasi-Armendariz rings. In this paper, we give a sufficient condition for a ring R and a monoid M such that the monoid ring R[M] is quasi-Armendariz. As a consequence we show that if R is a right APP-ring, then R[x]=(xn) and hence the trivial extension T(R,R) are quasi-Armendariz. They allow the construction of rings with a non-zero nilpotent ideal of arbitrary index of nilpotency which are quasi-Armendariz.

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NASR-ISFAHANI, A. R. "REVERSIBLE SKEW GENERALIZED POWER SERIES RINGS." Bulletin of the Australian Mathematical Society 84, no. 3 (July 21, 2011): 455–57. http://dx.doi.org/10.1017/s0004972711002450.

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AbstractIn this note we show that there exist a semiprime ring R, a strictly ordered artinian, narrow, unique product monoid (S,≤) and a monoid homomorphism ω:S⟶End(R) such that the skew generalized power series ring R[[S,ω]] is semicommutative but R[[S,ω]] is not reversible. This answers a question posed in Marks et al. [‘A unified approach to various generalizations of Armendariz rings’, Bull. Aust. Math. Soc.81 (2010), 361–397].

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27

ALHEVAZ, A., and D. KIANI. "McCOY PROPERTY OF SKEW LAURENT POLYNOMIALS AND POWER SERIES RINGS." Journal of Algebra and Its Applications 13, no. 02 (October 10, 2013): 1350083. http://dx.doi.org/10.1142/s0219498813500837.

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One of the important properties of commutative rings, proved by McCoy [Remarks on divisors of zero, Amer. Math. Monthly49(5) (1942) 286–295], is that if two nonzero polynomials annihilate each other over a commutative ring then each polynomial has a nonzero annihilator in the base ring. Nielsen [Semi-commutativity and the McCoy condition, J. Algebra298(1) (2006) 134–141] generalizes this property to non-commutative rings. Let M be a monoid and σ be an automorphism of a ring R. For the continuation of McCoy property of non-commutative rings, in this paper, we extend the McCoy's theorem to skew Laurent power series ring R[[x, x-1; σ]] and skew monoid ring R * M over general non-commutative rings. Constructing various examples, we classify how these skew versions of McCoy property behaves under various ring extensions. Moreover, we investigate relations between these properties and other standard ring-theoretic properties such as zip rings and rings with Property (A). As a consequence we extend and unify several known results related to McCoy rings.

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Anderson, David F., and John D. LaGrange. "Abian's poset and the ordered monoid of annihilator classes in a reduced commutative ring." Journal of Algebra and Its Applications 13, no. 08 (June 24, 2014): 1450070. http://dx.doi.org/10.1142/s0219498814500704.

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Let R be a reduced commutative ring with 1 ≠ 0. Then R is a partially ordered set under the Abian order defined by x ≤ y if and only if xy = x2. Let RE be the set of equivalence classes for the equivalence relation on R given by x ~ y if and only if ann R(x) = ann R(y). Then RE is a commutative Boolean monoid with multiplication [x][y] = [xy] and is thus partially ordered by [x] ≤ [y] if and only if [xy] = [x]. In this paper, we study R and RE as both monoids and partially ordered sets. We are particularly interested in when RE can be embedded in R as either a monoid or a partially ordered set.

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SHAH, TARIQ, and ANTONIO APARECIDO DE ANDRADE. "CYCLIC CODES THROUGH $B[X;\frac{a}{b}{\mathbb Z}_{0}]$, WITH $\frac{a}{b}\in {\mathbb Q}^{+}$ AND b = a+1, AND ENCODING." Discrete Mathematics, Algorithms and Applications 04, no. 04 (December 2012): 1250059. http://dx.doi.org/10.1142/s1793830912500590.

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Let B[X; S] be a monoid ring with any fixed finite unitary commutative ring B and [Formula: see text] is the monoid S such that b = a + 1, where a is any positive integer. In this paper we constructed cyclic codes, BCH codes, alternant codes, Goppa codes, Srivastava codes through monoid ring [Formula: see text]. For a = 1, almost all the results contained in [16] stands as a very particular case of this study.

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MARKS, GREG, RYSZARD MAZUREK, and MICHAŁ ZIEMBOWSKI. "A UNIFIED APPROACH TO VARIOUS GENERALIZATIONS OF ARMENDARIZ RINGS." Bulletin of the Australian Mathematical Society 81, no. 3 (February 23, 2010): 361–97. http://dx.doi.org/10.1017/s0004972709001178.

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AbstractLet R be a ring, S a strictly ordered monoid, and ω:S→End(R) a monoid homomorphism. The skew generalized power series ring R[[S,ω]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal’cev–Neumann Laurent series rings. We study the (S,ω)-Armendariz condition on R, a generalization of the standard Armendariz condition from polynomials to skew generalized power series. We resolve the structure of (S,ω)-Armendariz rings and obtain various necessary or sufficient conditions for a ring to be (S,ω)-Armendariz, unifying and generalizing a number of known Armendariz-like conditions in the aforementioned special cases. As particular cases of our general results we obtain several new theorems on the Armendariz condition; for example, left uniserial rings are Armendariz. We also characterize when a skew generalized power series ring is reduced or semicommutative, and we obtain partial characterizations for it to be reversible or 2-primal.

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31

Rugayah, Siti, Ahmad Faisol, and Fitriani Fitriani. "MATRIKS ATAS RING DERET PANGKAT TERGENERALISASI MIRING." BAREKENG: Jurnal Ilmu Matematika dan Terapan 15, no. 1 (March 1, 2021): 157–66. http://dx.doi.org/10.30598/barekengvol15iss1pp157-166.

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Let R be a ring with unit elements, strictly ordered monoids, and a monoid homomorphism. Formed , which is a set of all functions from S to R with are Artin and narrow. With the operation of the sum of functions and convolution multiplication, is a ring, from now on referred to as the Skew Generalized Power Series Ring (SGPSR). In this paper, the set of all matrices over SGPSR will be constructed. Furthermore, it will be shown that this set is a ring with the addition and multiplication matrix operations. Moreover, we will construct the ideal of ring matrix over SGPSR and investigate this ideal's properties.

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32

Tarizadeh, Abolfazl. "A fresh look into monoid rings and formal power series rings." Journal of Algebra and Its Applications 19, no. 01 (January 23, 2019): 2050003. http://dx.doi.org/10.1142/s0219498820500036.

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In this paper, the ring of polynomials is studied in a systematic way through the theory of monoid rings. As a consequence, this study provides canonical approaches in order to find easy and rigorous proofs and methods for many facts on polynomials and formal power series; some of them as sample are treated in this paper. Besides the universal properties of the monoid rings and polynomial rings, a universal property for the formal power series rings is also established.

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ZHONGKUI, LIU, and YANG XIAOYAN. "ON ANNIHILATOR IDEALS OF SKEW MONOID RINGS." Glasgow Mathematical Journal 52, no. 1 (December 4, 2009): 161–68. http://dx.doi.org/10.1017/s0017089509990255.

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AbstractA ring R is called a left APP-ring if the left annihilator lR(Ra) is pure as a left ideal of R for every a ∈ R; R is called (left principally) quasi-Baer if the left annihilator of every (principal) left ideal of R is generated by an idempotent. Let R be a ring and M an ordered monoid. Assume that there is a monoid homomorphism φ: M ⟶ Aut(R). We give a necessary and sufficient condition for the skew monoid ring (induced by φ) to be left APP (left principally quasi-Baer, quasi-Baer, respectively).

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34

Matsuda, Ryuki. "Note on integral closures of semigroup rings." Tamkang Journal of Mathematics 31, no. 2 (June 30, 2000): 137–44. http://dx.doi.org/10.5556/j.tkjm.31.2000.405.

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Let $S$ be a subsemigroup which contains 0 of a torsion-free abelian (additive) group. Then $S$ is called a grading monoid (or a $g$-monoid). The group $ \{s-s'|s,s'\in S\}$ is called the quotient group of $S$, and is denored by $q(S)$. Let $R$ be a commutative ring. The total quotient ring of $R$ is denoted by $q(R)$. Throught the paper, we assume that a $g$-monoid properly contains $ \{0\}$. A commutative ring is called a ring, and a non-zero-divisor of a ring is called a regular element of the ring. We consider integral elements over the semigroup ring $ R[X;S]$ of $S$ over $R$. Let $S$ be a $g$-monoid with quotient group $G$. If $ n\alpha\in S$ for an element $ \alpha$ of $G$ and a natural number $n$ implies $ \alpha\in S$, then $S$ is called an integrally closed semigroup. We know the following fact: ${\bf Theorem~1}$ ([G2, Corollary 12.11]). Let $D$ be an integral domain and $S$ a $g$-monoid. Then $D[X;S]$ is integrally closed if and only if $D$ is an integrally closed domain and $S$ is an integrally closed semigroup. Let $R$ be a ring. In this paper, we show that conditions for $R[X;S]$ to be integrally closed reduce to conditions for the polynomial ring of an indeterminate over a reduced total quotient ring to be integrally closed (Theorem 15). Clearly the quotient field of an integral domain is a von Neumann regular ring. Assume that $q(R)$ is a von Neumann regular ring. We show that $R[X;S]$ is integrally closed if and only if $R$ is integrally closed and $S$ is integrally closed (Theorem 20). Let $G$ be a $g$-monoid which is a group. If $R$ is a subring of the ring $T$ which is integrally closed in $T$, we show that $R[X;G]$ is integrally closed in $T[X;S]$ (Theorem 13). Finally, let $S$ be sub-$g$-monoid of a totally ordered abelian group. Let $R$ be a subring of the ring $T$ which is integrally closed in $T$. If $g$ and $h$ are elements of $T[X;S]$ with $h$ monic and $gh\in R[X;S]$, we show that $g\in R[X;S]$ (Theorem 24).

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Paykan, Kamal, and Ahmad Moussavi. "Nilpotent elements and nil-Armendariz property of skew generalized power series rings." Asian-European Journal of Mathematics 10, no. 02 (August 2, 2016): 1750034. http://dx.doi.org/10.1142/s1793557117500346.

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Let [Formula: see text] be a ring, [Formula: see text] a strictly ordered monoid, and [Formula: see text] a monoid homomorphism. The skew generalized power series ring [Formula: see text] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal’cev–Neumann Laurent series rings. In this paper, we introduce and study the [Formula: see text]-nil-Armendariz condition on [Formula: see text], a generalization of the standard nil-Armendariz condition from polynomials to skew generalized power series. We resolve the structure of [Formula: see text]-nil-Armendariz rings and obtain various necessary or sufficient conditions for a ring to be [Formula: see text]-nil-Armendariz. The [Formula: see text]-nil-Armendariz condition is connected to the question of whether or not a skew generalized power series ring [Formula: see text] over a nil ring [Formula: see text] is nil, which is related to a question of Amitsur [Algebras over infinite fields, Proc. Amer. Math. Soc. 7 (1956) 35–48]. As particular cases of our general results we obtain several new theorems on the nil-Armendariz condition. Our results extend and unify many existing results.

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Paykan, K., and A. Moussavi. "Quasi-Armendariz generalized power series rings." Journal of Algebra and Its Applications 15, no. 05 (March 30, 2016): 1650086. http://dx.doi.org/10.1142/s0219498816500869.

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Let [Formula: see text] be a ring, [Formula: see text] a strictly ordered monoid and [Formula: see text] a monoid homomorphism. The skew generalized power series ring [Formula: see text] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal’cev–Neumann Laurent series rings. We initiate the study of the [Formula: see text]-quasi-Armendariz condition on [Formula: see text], a generalization of the standard quasi-Armendariz condition from polynomials to skew generalized power series. The class of quasi-Armendariz rings includes semiprime rings, Armendariz rings, right (left) p.q.-Baer rings and right (left) PP rings. The [Formula: see text]-quasi-Armendariz rings are closed under direct sums, upper triangular matrix rings, full matrix rings and Morita invariance. The [Formula: see text] formal upper triangular matrix rings of this class are characterized. We conclude some characterizations for a skew generalized power series ring to be semiprime, quasi-Baer, generalized quasi-Baer, primary, nilary, reflexive, ideal-symmetric and left AIP. Examples to illustrate and delimit the theory are provided.

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37

Hashemi, E., M. Yazdanfar, and A. Alhevaz. "On generalized power series rings with some restrictions on zero-divisors." Journal of Algebra and Its Applications 17, no. 03 (February 5, 2018): 1850040. http://dx.doi.org/10.1142/s0219498818500408.

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Let [Formula: see text] be a ring and [Formula: see text] a strictly ordered monoid. The construction of generalized power series ring [Formula: see text] generalizes some ring constructions such as polynomial rings, group rings, power series rings and Mal’cev–Neumann construction. In this paper, for a reversible right Noetherian ring [Formula: see text] and a m.a.n.u.p. monoid [Formula: see text], it is shown that (i) [Formula: see text] is power-serieswise [Formula: see text]-McCoy, (ii) [Formula: see text] have Property (A), (iii) [Formula: see text] is right zip if and only if [Formula: see text] is right zip, (iv) [Formula: see text] is strongly AB if and only if [Formula: see text] is strongly AB. Also we study the interplay between ring-theoretical properties of a generalized power series ring [Formula: see text] and the graph-theoretical properties of its undirected zero divisor graph of [Formula: see text]. A complete characterization for the possible diameters [Formula: see text] is given exclusively in terms of the ideals of [Formula: see text]. Also, we present some examples to show that the assumption “R is right Noetherian” in our main results is not superfluous.

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38

Hirano, Yasuyuki. "ON ORDERED MONOID RINGS OVER A QUASI-BAER RING." Communications in Algebra 29, no. 5 (April 30, 2001): 2089–95. http://dx.doi.org/10.1081/agb-100002171.

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39

Paykan, Kamal, and Ahmad Moussavi. "McCoy property and nilpotent elements of skew generalized power series rings." Journal of Algebra and Its Applications 16, no. 10 (September 20, 2017): 1750183. http://dx.doi.org/10.1142/s0219498817501833.

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Let [Formula: see text] be a ring, [Formula: see text] a strictly ordered monoid and [Formula: see text] a monoid homomorphism. The skew generalized power series ring [Formula: see text] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal’cev–Neumann Laurent series rings. In this paper, we consider the problem of determining when [Formula: see text] is nilpotent in [Formula: see text]. We study various annihilator properties and a variety of conditions and related properties that the skew generalized power series [Formula: see text] inherits from [Formula: see text]. We also introduce and study the [Formula: see text]-McCoy condition on [Formula: see text], a generalization of the standard McCoy condition from polynomials to skew generalized power series. We resolve the structure of [Formula: see text]-McCoy rings and obtain various necessary or sufficient conditions for a ring to be [Formula: see text]-McCoy. As particular cases of our general results we obtain several new theorems on the McCoy condition. Moreover various examples of [Formula: see text]-McCoy rings are provided.

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40

Hashemi, E., A. AS Estaji, and M. Ziembowski. "Answers to Some Questions Concerning Rings with Property (A)." Proceedings of the Edinburgh Mathematical Society 60, no. 3 (January 31, 2017): 651–64. http://dx.doi.org/10.1017/s0013091516000407.

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AbstractA ring R has right property (A) whenever a finitely generated two-sided ideal of R consisting entirely of left zero-divisors has a non-zero right annihilator. As the main result of this paper we give answers to two questions related to property (A), raised by Hong et al. One of the questions has a positive answer and we obtain it as a simple conclusion of the fact that if R is a right duo ring and M is a u.p.-monoid (unique product monoid), then R is right M-McCoy and the monoid ring R[M] has right property (A). The second question has a negative answer and we demonstrate this by constructing a suitable example.

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HONG, CHAN YONG, NAM KYUN KIM, and YANG LEE. "EXTENSIONS OF McCOY'S THEOREM." Glasgow Mathematical Journal 52, no. 1 (December 4, 2009): 155–59. http://dx.doi.org/10.1017/s0017089509990243.

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AbstractMcCoy proved that for a right ideal A of S = R[x1, . . ., xk] over a ring R, if rS(A) ≠ 0 then rR(A) ≠ 0. We extend the result to the Ore extensions, the skew monoid rings and the skew power series rings over non-commutative rings and so on.

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42

KUBER, AMIT. "On the Grothendieck ring of varieties." Mathematical Proceedings of the Cambridge Philosophical Society 158, no. 3 (January 18, 2015): 477–86. http://dx.doi.org/10.1017/s0305004115000079.

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AbstractLet K0(Vark) denote the Grothendieck ring of k-varieties over an algebraically closed field k. Larsen and Lunts asked if two k-varieties having the same class in K0(Vark) are piecewise isomorphic. Gromov asked if a birational self-map of a k-variety can be extended to a piecewise automorphism. We show that these two questions are equivalent over any algebraically closed field. If these two questions admit a positive answer, then we prove that its underlying abelian group is a free abelian group and that the associated graded ring of the Grothendieck ring is the monoid ring $\mathbb{Z}$[$\mathfrak{B}$] where $\mathfrak{B}$ denotes the multiplicative monoid of birational equivalence classes of irreducible k-varieties.

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43

Gould, Victoria, Miklós Hartmann, and Nik Ruškuc. "Free Monoids are Coherent." Proceedings of the Edinburgh Mathematical Society 60, no. 1 (June 15, 2016): 127–31. http://dx.doi.org/10.1017/s0013091516000079.

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AbstractA monoid S is said to be right coherent if every finitely generated subact of every finitely presented right S-act is finitely presented. Left coherency is defined dually and S is coherent if it is both right and left coherent. These notions are analogous to those for a ring R (where, of course, S-acts are replaced by R-modules). Choo et al. have shown that free rings are coherent. In this paper we prove that, correspondingly, any free monoid is coherent, thus answering a question posed by Gould in 1992.

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44

SHAH, TARIQ, and ANTONIO APARECIDO DE ANDRADE. "CYCLIC CODES THROUGH B[X], $B[X;\frac{1}{kp}Z_{0}]$ AND $B[X;\frac{1}{p^{k}}Z_{0}]$: A COMPARISON." Journal of Algebra and Its Applications 11, no. 04 (July 31, 2012): 1250078. http://dx.doi.org/10.1142/s0219498812500788.

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It is very well known that algebraic structures have valuable applications in the theory of error-correcting codes. Blake [Codes over certain rings, Inform. and Control 20 (1972) 396–404] has constructed cyclic codes over ℤm and in [Codes over integer residue rings, Inform. and Control 29 (1975), 295–300] derived parity check-matrices for these codes. In [Linear codes over finite rings, Tend. Math. Appl. Comput. 6(2) (2005) 207–217]. Andrade and Palazzo present a construction technique of cyclic, BCH, alternant, Goppa and Srivastava codes over a local finite ring B. However, in [Encoding through generalized polynomial codes, Comput. Appl. Math. 30(2) (2011) 1–18] and [Constructions of codes through semigroup ring [Formula: see text] and encoding, Comput. Math. Appl. 62 (2011) 1645–1654], Shah et al. extend this technique of constructing linear codes over a finite local ring B via monoid rings [Formula: see text], where p = 2 and k = 1, 2, respectively, instead of the polynomial ring B[X]. In this paper, we construct these codes through the monoid ring [Formula: see text], where p = 2 and k = 1, 2, 3. Moreover, we also strengthen and generalize the results of [Encoding through generalized polynomial codes, Comput. Appl. Math.30(2) (2011) 1–18] and [Constructions of codes through semigroup ring [Formula: see text]] and [Encoding, Comput. Math. Appl.62 (2011) 1645–1654] to the case of k = 3.

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45

Tariq Shah. "Irreducible Generalized Polynomials in a Monoid Ring." Journal of Advanced Research in Pure Mathematics 3, no. 4 (November 1, 2011): 24–32. http://dx.doi.org/10.5373/jarpm.649.112410.

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46

Schwiebert, Ryan C. "The radical-annihilator monoid of a ring." Communications in Algebra 45, no. 4 (October 7, 2016): 1601–17. http://dx.doi.org/10.1080/00927872.2016.1222401.

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47

Faisol, Ahmad, and Fitriani Fitriani. "The Sufficient Conditions for M[[S,w]] to be T[[S,w]]-Noetherian R[[S,w]]-module." Al-Jabar : Jurnal Pendidikan Matematika 10, no. 2 (December 18, 2019): 285–92. http://dx.doi.org/10.24042/ajpm.v10i2.5042.

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In this paper, we investigate the sufficient conditions for T[[S,w]] to be a multiplicative subset of skew generalized power series ring R[[S,w]], where R is a ring, T Í R a multiplicative set, (S,≤) a strictly ordered monoid, and w : S®End(R) a monoid homomorphism. Furthermore, we obtain sufficient conditions for skew generalized power series module M[[S,w]] to be a T[[S,w]]-Noetherian R[[S,w]]-module, where M is an R-module.

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48

NASR-ISFAHANI, A. R. "RADICALS OF SKEW GENERALIZED POWER SERIES RINGS." Journal of Algebra and Its Applications 12, no. 01 (December 13, 2012): 1250129. http://dx.doi.org/10.1142/s0219498812501290.

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Let R be a ring, (S, ≤) a strictly ordered monoid and ω : S → End (R) a monoid homomorphism. In this note for a (S, ω)-Armendariz ring R we study some properties of skew generalized power series ring R[[S, ω]]. In particular, among other results, we show that for a S-compatible (S, ω)-Armendariz ring R, α(R[[S, ω]]) = α(R)[[S, ω]] = Ni ℓ*(R)[[S, ω]], where α is a radical in a class of radicals which includes the Wedderburn, lower nil, Levitzky and upper nil radicals. We also show that several properties, including the symmetric, reversible, ZCn, zip and 2-primal property, transfer between R and the skew generalized power series ring R[[S, ω]], in case R is S-compatible (S, ω)-Armendariz.

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Liu, Zhongkui. "PF-rings of generalised power series." Bulletin of the Australian Mathematical Society 57, no. 3 (June 1998): 427–32. http://dx.doi.org/10.1017/s0004972700031841.

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Let R be a commutative ring and (S, ≤) a strictly ordered monoid which satisfies the condition that 0 ≤ s for every s ∈ S. We show that the generalised power series ring [[RS ≤]] is a PF-ring if and only if R is a PF-ring.

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50

Lopatkin, Viktor. "Cohomology rings of the plactic monoid algebra via a Gröbner–Shirshov basis." Journal of Algebra and Its Applications 15, no. 05 (March 30, 2016): 1650082. http://dx.doi.org/10.1142/s0219498816500821.

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In this paper, we calculate the cohomology ring [Formula: see text] and the Hochschild cohomology ring of the plactic monoid algebra [Formula: see text] via the Anick resolution using a Gröbner–Shirshov basis.

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