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Journal articles on the topic 'Nilpotentní ideál'

Author: Grafiati

Published: 26 July 2021

Last updated: 10 February 2022

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1

Marques-Smith, M. Paula O., and R. P. Sullivan. "The ideal structure of nilpotent-generated transformation semigroups." Bulletin of the Australian Mathematical Society 60, no. 2 (October 1999): 303–18. http://dx.doi.org/10.1017/s0004972700036418.

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In 1987 Sullivan determined the elements of the semigroup N(X) generated by all nilpotent partial transformations of an infinite set X; and later in 1997 he studied subsemigroups of N(X) defined by restricting the index of the nilpotents and the cardinality of the set. Here, we describe the ideals and Green's relations on such semigroups, like Reynolds and Sullivan did in 1985 for the semigroup generated by all idempotent total transformations of X. We then use this information to describe the congruences on certain Rees factor semigroups and to construct families of congruence-free semigroups with interesting algebraic properties. We also study analogous questions for X finite and for one-to-one partial transformations.

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2

Yaqub, Adil. "Weakly periodic-like rings and commutativity." Studia Scientiarum Mathematicarum Hungarica 43, no. 3 (September 1, 2006): 275–84. http://dx.doi.org/10.1556/sscmath.43.2006.3.1.

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A ring R is called periodic if, for every x in R, there exist distinct positive integers m and n such that xm=xn. An element x is called potent if xk=x for some integer k≯1. A ring R is called weakly periodic if every x in R can bewritten in the form x=a+b for some nilpotent element a and some potent element b. A ring R is called weakly periodic-like if every x in R which is not in the center of R can be written in the form x=a+b, where a is nilpotent and b is potent. Our objective is to study the structure of weakly periodic-like rings, with particular emphasis on conditions which yield such rings commutative, or conditions which render the nilpotents N as an ideal of R and R/N as commutative.

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3

Abu-Khuzam, Hazar, and Adil Yaqub. "Structure of Certain Periodic Rings." Canadian Mathematical Bulletin 28, no. 1 (March 1, 1985): 120–23. http://dx.doi.org/10.4153/cmb-1985-014-9.

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AbstractLet R be a periodic ring, N the set of nilpotents, and D the set of right zero divisors of R. Suppose that (i) N is commutative, and (ii) every x in R can be uniquely written in the form x = e + a, where e2 = e and a ∊ N. Then N is an ideal in R and R/N is a Boolean ring. If (i) is satisfied but (ii) is now assumed to hold merely for those elements x ∊ D, and if 1 ∊ R, then N is still an ideal in R and R/N is a subdirect sum of fields. It is further shown that if (i) is satisfied but (ii) is replaced by: "every right zero divisor is either nilpotent or idempotent," and if 1 ∊ R, then N is still an ideal in R and R/N is either a Boolean ring or a field.

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4

Mohammadzadeh, E., G. Muhiuddin, J. Zhan, and R. A. Borzooei. "Nilpotent fuzzy lie ideals." Journal of Intelligent & Fuzzy Systems 39, no. 3 (October 7, 2020): 4071–79. http://dx.doi.org/10.3233/jifs-200211.

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In this paper, we introduce a new definition for nilpotent fuzzy Lie ideal, which is a well-defined extension of nilpotent Lie ideal in Lie algebras, and we name it a good nilpotent fuzzy Lie ideal. Then we prove that a Lie algebra is nilpotent if and only if any fuzzy Lie ideal of it, is a good nilpotent fuzzy Lie ideal. In particular, we construct a nilpotent Lie algebra via a good nilpotent fuzzy Lie ideal. Also, we prove that with some conditions, every good nilpotent fuzzy Lie ideal is finite. Finally, we define an Engel fuzzy Lie ideal, and we show that every Engel fuzzy Lie ideal of a finite Lie algebra is a good nilpotent fuzzy Lie ideal. We think that these notions could be useful to solve some problems of Lie algebras with nilpotent fuzzy Lie ideals.

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5

Woods, William. "On the structure of virtually nilpotent compact p-adic analytic groups." Journal of Group Theory 21, no. 1 (January 1, 2018): 165–88. http://dx.doi.org/10.1515/jgth-2017-0017.

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AbstractLetGbe a compactp-adic analytic group. We recall the well-understood finite radical{\Delta^{+}}and FC-centre Δ, and introduce ap-adic analogue of Roseblade’s subgroup{\mathrm{nio}(G)}, the unique largest orbitally sound open normal subgroup ofG. Further, whenGis nilpotent-by-finite, we introduce the finite-by-(nilpotentp-valuable) radical{\mathbf{FN}_{p}(G)}, an open characteristic subgroup ofGcontained in{\mathrm{nio}(G)}. By relating the already well-known theory of isolators with Lazard’s notion ofp-saturations, we introduce the isolated lower central (resp. isolated derived) series of a nilpotent (resp. soluble)p-valuable group, and use this to study the conjugation action of{\mathrm{nio}(G)}on{\mathbf{FN}_{p}(G)}. We emerge with a structure theorem forG,1\leq\Delta^{+}\leq\Delta\leq\mathbf{FN}_{p}(G)\leq\mathrm{nio}(G)\leq G,in which the various quotients of this series of groups are well understood. This sheds light on the ideal structure of the Iwasawa algebras (i.e. the completed group ringskG) of such groups, and will be used in future work to study the prime ideals of these rings.

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6

PANYUSHEV, DMITRI I., and OKSANA S. YAKIMOVA. "NILPOTENT SUBSPACES AND NILPOTENT ORBITS." Journal of the Australian Mathematical Society 106, no. 1 (May 30, 2018): 104–26. http://dx.doi.org/10.1017/s1446788718000071.

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Let $G$ be a semisimple complex algebraic group with Lie algebra $\mathfrak{g}$. For a nilpotent $G$-orbit ${\mathcal{O}}\subset \mathfrak{g}$, let $d_{{\mathcal{O}}}$ denote the maximal dimension of a subspace $V\subset \mathfrak{g}$ that is contained in the closure of ${\mathcal{O}}$. In this note, we prove that $d_{{\mathcal{O}}}\leq {\textstyle \frac{1}{2}}\dim {\mathcal{O}}$ and this upper bound is attained if and only if ${\mathcal{O}}$ is a Richardson orbit. Furthermore, if $V$ is $B$-stable and $\dim V={\textstyle \frac{1}{2}}\dim {\mathcal{O}}$, then $V$ is the nilradical of a polarisation of ${\mathcal{O}}$. Every nilpotent orbit closure has a distinguished $B$-stable subspace constructed via an $\mathfrak{sl}_{2}$-triple, which is called the Dynkin ideal. We then characterise the nilpotent orbits ${\mathcal{O}}$ such that the Dynkin ideal (1) has the minimal dimension among all $B$-stable subspaces $\mathfrak{c}$ such that $\mathfrak{c}\cap {\mathcal{O}}$ is dense in $\mathfrak{c}$, or (2) is the only $B$-stable subspace $\mathfrak{c}$ such that $\mathfrak{c}\cap {\mathcal{O}}$ is dense in $\mathfrak{c}$.

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7

Eslami, Esfandiar, and Patrick Stewart. "Two-sided essential nilpotence." International Journal of Mathematics and Mathematical Sciences 15, no. 2 (1992): 351–54. http://dx.doi.org/10.1155/s0161171292000449.

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An idealIof a ringAis essentially nilpotent ifIcontains a nilpotent idealNofAsuch thatJ⋂N≠0wheneverJis a nonzero ideal ofAcontained inI. We show that each ringAhas a unique largest essentially nilpotent idealEN(A). We study the properties ofEN(A)and, in particular, we investigate how this ideal behaves with respect to related rings.

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8

JESPERS, E., and M. H. SHAHZAMANIAN. "A DESCRIPTION OF A CLASS OF FINITE SEMIGROUPS THAT ARE NEAR TO BEING MAL'CEV NILPOTENT." Journal of Algebra and Its Applications 12, no. 05 (May 7, 2013): 1250221. http://dx.doi.org/10.1142/s0219498812502210.

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In this paper we continue the investigation on the algebraic structure of a finite semigroup S that is determined by its associated upper non-nilpotent graph [Formula: see text]. The vertices of this graph are the elements of S and two vertices are adjacent if they generate a semigroup that is not nilpotent (in the sense of Mal'cev). We introduce a class of semigroups in which the Mal'cev nilpotent property lifts through ideal chains. We call this the class of pseudo-nilpotent semigroups. The definition is such that the global information that a semigroup is not nilpotent induces local information, i.e. some two-generated subsemigroups are not nilpotent. It turns out that a finite monoid (in particular, a finite group) is pseudo-nilpotent if and only if it is nilpotent. Our main result is a description of pseudo-nilpotent finite semigroups S in terms of their associated graph [Formula: see text]. In particular, S has a largest nilpotent ideal, say K, and S/K is a 0-disjoint union of its connected components (adjoined with a zero) with at least two elements.

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9

Röhl, Frank. "A Remark on the Loewy-Series of Certain Hopf Algebras." Canadian Mathematical Bulletin 32, no. 2 (June 1, 1989): 190–93. http://dx.doi.org/10.4153/cmb-1989-028-6.

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AbstractAn easy proof will be given to show that for finite dimensional Hopf-algebras with nilpotent augmentation ideal over the field of p elements, the upper and lower Loewy-series coincide. In particular, this holds for the restricted universal envelope of nilpotent Lie-p-algebras with nilpotent p-map.

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10

Artemovych, Orest D. "Associative rings in which nilpotents form an ideal." Studia Scientiarum Mathematicarum Hungarica 56, no. 2 (June 2019): 177–84. http://dx.doi.org/10.1556/012.2019.56.2.1428.

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Abstract It is shown that if N(R) is a Lie ideal of R (respectively Jordan ideal and R is 2-torsion-free), then N(R) is an ideal. Also, it is presented a characterization of Noetherian NR rings with central idempotents (respectively with the commutative set of nilpotent elements, the Abelian unit group, the commutative commutator set).

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11

Johnston, Carolyn Pfeffer. "Primitive Ideal Spaces of Discrete Rational Nilpotent Groups." American Journal of Mathematics 117, no. 2 (April 1995): 323. http://dx.doi.org/10.2307/2374917.

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12

Carey, A., and William Moran. "Nilpotent groups with $T_{1}$ primitive ideal spaces." Studia Mathematica 83, no. 1 (1986): 25–32. http://dx.doi.org/10.4064/sm-83-1-25-32.

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13

Qu, Yinchun, and Junchao Wei. "Rings whose nilpotent elements form a Lie ideal." Studia Scientiarum Mathematicarum Hungarica 51, no. 2 (June 1, 2014): 271–84. http://dx.doi.org/10.1556/sscmath.51.2014.2.1279.

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A ring R is called NLI (rings whose nilpotent elements form a Lie ideal) if for each a ∈ N(R) and b ∈ R, ab − ba ∈ N(R). Clearly, NI rings are NLI. In this note, many properties of NLI rings are studied. The main results we obtain are the following: (1) NLI rings are directly finite and left min-abel; (2) If R is a NLI ring, then (a) R is a strongly regular ring if and only if R is a Von Neumann regular ring; (b) R is (weakly) exchange if and only if R is (weakly) clean; (c) R is a reduced ring if and only if R is a n-regular ring; (3) If R is a NLI left MC2 ring whose singular simple left modules are Wnil-injective, then R is reduced.

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14

Voll, Christopher. "IDEAL ZETA FUNCTIONS ASSOCIATED TO A FAMILY OF CLASS-2-NILPOTENT LIE RINGS." Quarterly Journal of Mathematics 71, no. 3 (June 17, 2020): 959–80. http://dx.doi.org/10.1093/qmathj/haaa010.

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Abstract We produce explicit formulae for various ideal zeta functions associated to the members of an infinite family of class-$2$-nilpotent Lie rings, introduced in M. N. Berman, B. Klopsch and U. Onn (A family of class-2 nilpotent groups, their automorphisms and pro-isomorphic zeta functions, Math. Z. 290 (2018), 909935), in terms of Igusa functions. As corollaries we obtain information about analytic properties of global ideal zeta functions, local functional equations, topological, reduced and graded ideal zeta functions, as well as representation zeta functions for the unipotent group schemes associated to the Lie rings in question.

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15

Goswami, Nabanita, and Helen K. Saikia. "On nilpotency of the right singular ideal of semiring." Boletim da Sociedade Paranaense de Matemática 37, no. 2 (April 23, 2017): 123–27. http://dx.doi.org/10.5269/bspm.v37i2.34308.

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We introduce the concept of nilpotency of the right singular ideal of a semiring. We discuss some properties of such nilpotency and singular ideals. We show that the right singular ideal of a semiring with a.c.c. for right annihilators, is nilpotent.

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16

Johnson, Ben, and Eric Sommers. "Equations for Some Nilpotent Varieties." International Mathematics Research Notices 2020, no. 14 (July 2, 2018): 4433–64. http://dx.doi.org/10.1093/imrn/rny132.

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AbstractLet ${\mathcal{O}}$ be a Richardson nilpotent orbit in a simple Lie algebra $\mathfrak{g}$ of rank $n$ over $\mathbb C$, induced from a Levi subalgebra whose $s$ simple roots are orthogonal, short roots. The main result of the paper is a description of a minimal set of generators of the ideal defining $\overline{\mathcal{O}}$ in $S \mathfrak{g}^{\ast }$. In such cases, the ideal is generated by bases of either one or two copies of the representation whose highest weight is the dominant short root, along with $n-s$ fundamental invariants of $S \mathfrak{g}^{\ast }$. This extends Broer’s result for the subregular nilpotent orbit, which is the case of $s=1$. Along the way we give another proof of Broer’s result that $\overline{\mathcal{O}}$ is normal. We also prove a result relating a property of the invariants of $S \mathfrak{g}^{\ast }$ to the following question: when does a copy of the adjoint representation in $S \mathfrak{g}^{\ast }$ belong to the ideal in $S \mathfrak{g}^{\ast }$ generated by another copy of the adjoint representation together with the invariants of $S \mathfrak{g}^{\ast }$?

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17

Mauceri, Silvana, and Paola Misso. "Derivations on a Lie Ideal." Canadian Mathematical Bulletin 31, no. 3 (September 1, 1988): 280–86. http://dx.doi.org/10.4153/cmb-1988-041-2.

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AbstractIn this paper we prove the following result: let R be a prime ring with no non-zero nil left ideals whose characteristic is different from 2 and let U be a non central Lie ideal of R.If d ≠ 0 is a derivation of R such that d(u) is invertible or nilpotent for all u ∈ U, then either R is a division ring or R is the 2 X 2 matrices over a division ring. Moreover in the last case if the division ring is non commutative, then d is an inner derivation of R.

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18

Linckelmann, Markus. "On graded centres and block cohomology." Proceedings of the Edinburgh Mathematical Society 52, no. 2 (May 28, 2009): 489–514. http://dx.doi.org/10.1017/s0013091507001137.

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AbstractWe extend the group theoretic notions of transfer and stable elements to graded centres of triangulated categories. When applied to the centre Z*(Db(B) of the derived bounded category of a block algebra B we show that the block cohomology H*(B) is isomorphic to a quotient of a certain subalgebra of stable elements of Z*(Db(B)) by some nilpotent ideal, and that a quotient of Z*(Db(B)) by some nilpotent ideal is Noetherian over H*(B).

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19

Bell, Howard E., and Adil Yaqub. "Generalized periodic rings." International Journal of Mathematics and Mathematical Sciences 19, no. 1 (1996): 87–92. http://dx.doi.org/10.1155/s0161171296000130.

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LetRbe a ring, and letNandCdenote the set of nilpotents and the center ofR, respectively.Ris called generalized periodic if for everyx∈R\(N⋃C), there exist distinct positive integersm,nof opposite parity such thatxn−xm∈N⋂C. We prove that a generalized periodic ring always has the setNof nilpotents forming an ideal inR. We also consider some conditions which imply the commutativity of a generalized periodic ring.

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20

Roh, E. H., Y. B. Jun, S. Y. Kim, and W. H. Shim. "On properties of nil subsets in difference algebras." Tamkang Journal of Mathematics 32, no. 3 (September 30, 2001): 167–72. http://dx.doi.org/10.5556/j.tkjm.32.2001.371.

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In this paper, we introduce the concept of nil subsets by using nilpotent elements, and investigate some related properties. We show that a nil subset on a subalgebra (resp. (closed) ideal) is a subalgebra (resp. (closed) ideal). We also prove that in a nil algebra every ideal is a subalgebra.

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21

Mohammadzadeh, Elahe, and Rajab Ali Borzooei. "Engel, Nilpotent and Solvable BCI-algebras." Analele Universitatii "Ovidius" Constanta - Seria Matematica 27, no. 1 (March 1, 2019): 169–92. http://dx.doi.org/10.2478/auom-2019-0009.

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Abstract In this paper, we define the concepts of Engel, nilpotent and solvable BCI-algebras and investigate some of their properties. Specially, we prove that any BCK-algebra is a 2-Engel. Then we define the center of a BCI-algebra and prove that in a nilpotent BCI-algebra X, each minimal closed ideal of X is contained in the center of X. In addition, with some conditions, we show that every finite BCI-algebra is solvable. Finally, we investigate the relations among Engel, nilpotent and solvable BCI(BCK)-algebras.

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22

Jespers, Eric. "Special Principal Ideal Rings and Absolute Subretracts." Canadian Mathematical Bulletin 34, no. 3 (September 1, 1991): 364–67. http://dx.doi.org/10.4153/cmb-1991-058-6.

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AbstractA ring R is said to be an absolute subretract if for any ring S in the variety generated by R and for any ring monomorphism f from R into S, there exists a ring morphism g from S to R such that gf is the identity mapping. This concept, introduced by Gardner and Stewart, is a ring theoretic version of an injective notion in certain varieties investigated by Davey and Kovacs.Also recall that a special principal ideal ring is a local principal ring with nonzero nilpotent maximal ideal. In this paper (finite) special principal ideal rings that are absolute subretracts are studied.

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23

Armstrong, Grant F., and Stefan Sigg. "On the cohomology of a class of nilpotent Lie algebras." Bulletin of the Australian Mathematical Society 54, no. 3 (December 1996): 517–27. http://dx.doi.org/10.1017/s0004972700021936.

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Let g denote a finite dimensional nilpotent Lie algebra over ℂ containing an Abelian ideal a of codimension 1, with z ∈ g/a. We give a combinatorial description of the Betti numbers of g in terms of the Jordan decomposition induced by ad(z)|a. As an application we prove that the filiform-nilpotent Lie algebras arising in the case t = 1 have unimodal Betti numbers.

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24

Zubrilin, K. A. "On the largest nilpotent ideal in algebras satisfying Capelli identities." Sbornik: Mathematics 188, no. 8 (August 31, 1997): 1203–11. http://dx.doi.org/10.1070/sm1997v188n08abeh000253.

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25

Levasseur, T., S. P. Smith, and J. T. Stafford. "The minimal nilpotent orbit, the Joseph ideal, and differential operators." Journal of Algebra 116, no. 2 (August 1988): 480–501. http://dx.doi.org/10.1016/0021-8693(88)90231-1.

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26

Hejazian, S., and S. Talebi. "Derivations on Banach algebras." International Journal of Mathematics and Mathematical Sciences 2003, no. 28 (2003): 1803–6. http://dx.doi.org/10.1155/s0161171203209108.

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LetDbe a derivation on a Banach algebra; by using the operatorD2, we give necessary and sufficient conditions for the separating ideal ofDto be nilpotent. We also introduce an idealM(D)and apply it to find out more equivalent conditions for the continuity ofDand for nilpotency of its separating ideal.

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27

Aka, Menny, Emmanuel Breuillard, Lior Rosenzweig, and Nicolas de Saxcé. "Diophantine properties of nilpotent Lie groups." Compositio Mathematica 151, no. 6 (January 13, 2015): 1157–88. http://dx.doi.org/10.1112/s0010437x14007854.

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A finitely generated subgroup ${\rm\Gamma}$ of a real Lie group $G$ is said to be Diophantine if there is ${\it\beta}>0$ such that non-trivial elements in the word ball $B_{{\rm\Gamma}}(n)$ centered at $1\in {\rm\Gamma}$ never approach the identity of $G$ closer than $|B_{{\rm\Gamma}}(n)|^{-{\it\beta}}$. A Lie group $G$ is said to be Diophantine if for every $k\geqslant 1$ a random $k$-tuple in $G$ generates a Diophantine subgroup. Semi-simple Lie groups are conjectured to be Diophantine but very little is proven in this direction. We give a characterization of Diophantine nilpotent Lie groups in terms of the ideal of laws of their Lie algebra. In particular we show that nilpotent Lie groups of class at most $5$, or derived length at most $2$, as well as rational nilpotent Lie groups are Diophantine. We also find that there are non-Diophantine nilpotent and solvable (non-nilpotent) Lie groups.

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28

Buys, A., and G. K. Gerber. "Nil and s-prime Ω-groups." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 38, no. 2 (April 1985): 222–29. http://dx.doi.org/10.1017/s1446788700023089.

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AbstractThe concepts nilpotent element, s-prime ideal and s-semi-prime ideal are defined for Ω-groups. The class {G|G is a nil Ω-group} is a Kurosh-Amitsur radical class. The nil radical of an Ω-group coincides with the intersection of all the s-prime ideals. Furthermore an ideal P of G is an s-semi-prime ideal if and only if G/P has no non-zero nil ideals.

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29

Barnes, Donald W., and Daniel Groves. "The Wielandt Subalgebra of a Lie Algebra." Journal of the Australian Mathematical Society 74, no. 3 (June 2003): 313–30. http://dx.doi.org/10.1017/s1446788700003347.

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AbstractFollowing the analogy with group theory, we define the Wielandt subalgebra of a finite-dimensional Lie algebra to be the intersection of the normalisers of the subnormal subalgebras. In a non-zero algebra, this is a non-zero ideal if the ground field has characteristic 0 or if the derived algebra is nilpotent, allowing the definition of the Wielandt series. For a Lie algebra with nilpotent derived algebra, we obtain a bound for the derived length in terms of the Wielandt length and show this bound to be best possible. We also characterise the Lie algebras with nilpotent derived algebra and Wielandt length 2.

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30

Karparvar, Ali Mohammad, Babak Amini, Afshin Amini, and Habib Sharif. "Additive decomposition of ideals." Journal of Algebra and Its Applications 17, no. 05 (April 26, 2018): 1850085. http://dx.doi.org/10.1142/s0219498818500858.

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In this paper, we investigate decomposition of (one-sided) ideals of a unital ring [Formula: see text] as a sum of two (one-sided) ideals, each being idempotent, nil, nilpotent, T-nilpotent, or a direct summand of [Formula: see text]. Among other characterizations, we prove that in a polynomial identity ring every (one-sided) ideal is a sum of a nil (one-sided) ideal and an idempotent (one-sided) ideal if and only if the Jacobson radical [Formula: see text] of [Formula: see text] is nil and [Formula: see text] is von Neumann regular. As a special case, these conditions for a commutative ring [Formula: see text] are equivalent to [Formula: see text] having zero Krull dimension. While assuming Köthe’s conjecture in several occasions to be true, we also raise a question, the affirmative answer to which leads to the truth of the conjecture.

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31

Jayalakshmi and S. Madhavi Latha. "RIGHT NUCLEUS IN GENERALIZED RIGHT ALTERNATIVE RINGS." International Journal of Research -GRANTHAALAYAH 3, no. 1 (January 31, 2015): 1–12. http://dx.doi.org/10.29121/granthaalayah.v3.i1.2015.3044.

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Some properties of the right nucleus in generalized right alternative rings have been presented in this paper. In a generalized right alternative ring R which is finitely generated or free of locally nilpotent ideals, the right nucleus Nr equals the center C. Also, if R is prime and Nr ¹ C, then the associator ideal of R is locally nilpotent. Seong Nam [5] studied the properties of the right nucleus in right alternative algebra. He showed that if R is a prime right alternative algebra of char. ≠ 2 and Right nucleus Nr is not equal to the center C, then the associator ideal of R is locally nilpotent. But the problem arises when it come with the study of generalized right alternative ring as the ring dose not absorb the right alternative identity. In this paper we consider our ring to be generalized right alternative ring and try to prove the results of Seong Nam [5]. At the end of this paper we give an example to show that the generalized right alternative ring is not right alternative.

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Kaniuth, Eberhard, and William Moran. "THE GLIMM IDEAL SPACE OF A TWO-STEP NILPOTENT LOCALLY COMPACT GROUP." Proceedings of the Edinburgh Mathematical Society 44, no. 3 (October 2001): 505–26. http://dx.doi.org/10.1017/s0013091599001315.

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AbstractFor a two-step nilpotent locally compact group $G$, we determine the Glimm ideal space of the group $C^*$-algebra $C^*(G)$ and its topology. This leads to necessary and sufficient conditions for $C^*(G)$ to be quasi-standard. Moreover, some results about the Glimm classes of points in the primitive ideal space $\mathrm{Prim}(C^*(G))$ are obtained.AMS 2000 Mathematics subject classification: Primary 22D25. Secondary 22D10

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33

del Barco, Viviana. "On a spectral sequence for the cohomology of a nilpotent Lie algebra." Journal of Algebra and Its Applications 14, no. 01 (September 10, 2014): 1450078. http://dx.doi.org/10.1142/s0219498814500789.

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Given a nilpotent Lie algebra 𝔫 we construct a spectral sequence which is derived from a filtration of its Chevalley–Eilenberg differential complex and converges to the Lie algebra cohomology of 𝔫. The limit of this spectral sequence gives a grading for the Lie algebra cohomology, except for the cohomology groups of degree 0, 1, dim 𝔫 - 1 and dim 𝔫 as we shall prove. We describe the spectral sequence associated to a nilpotent Lie algebra which is a direct sum of two ideals, one of them of dimension one, in terms of the spectral sequence of the co-dimension one ideal. Also, we compute the spectral sequence corresponding to each real nilpotent Lie algebra of dimension less than or equal to six.

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34

Loke, Hung Yean, and Gordan Savin. "On the Maximal Primitive Ideal Corresponding to the Model Nilpotent Orbit." International Mathematics Research Notices 2012, no. 24 (2012): 5731–43. http://dx.doi.org/10.1093/imrn/rnr257.

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35

Baggett, L., and J. Packer. "The Primitive Ideal Space of Two-Step Nilpotent Group C*-Algebras." Journal of Functional Analysis 124, no. 2 (September 1994): 389–426. http://dx.doi.org/10.1006/jfan.1994.1112.

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36

Baggett, Lawrence W., Eberhard Kaniuth, and William Moran. "Primitive Ideal Spaces, Characters, and Kirillov Theory for Discrete Nilpotent Groups." Journal of Functional Analysis 150, no. 1 (October 1997): 175–203. http://dx.doi.org/10.1006/jfan.1997.3115.

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37

MA, LINQUAN, and PHAM HUNG QUY. "FROBENIUS ACTIONS ON LOCAL COHOMOLOGY MODULES AND DEFORMATION." Nagoya Mathematical Journal 232 (September 7, 2017): 55–75. http://dx.doi.org/10.1017/nmj.2017.20.

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Let $(R,\mathfrak{m})$ be a Noetherian local ring of characteristic $p>0$. We introduce and study $F$-full and $F$-anti-nilpotent singularities, both are defined in terms of the Frobenius actions on the local cohomology modules of $R$ supported at the maximal ideal. We prove that if $R/(x)$ is $F$-full or $F$-anti-nilpotent for a nonzero divisor $x\in R$, then so is $R$. We use these results to obtain new cases on the deformation of $F$-injectivity.

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38

Ludwig, J., and C. Molitor-Braun. "Exponential actions, orbits and their kernels." Bulletin of the Australian Mathematical Society 57, no. 3 (June 1998): 497–513. http://dx.doi.org/10.1017/s0004972700031919.

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Let be a nilpotent Lie algebra which is an exponential -module, being an exponential algebra of derivations of . Put = exp and = exp . If Ω is a closed orbit of * under the action of , then Ker is dense in Ker Ω for the topology of L1 () and the algebra Ker is nilpotent, where denotes the minimal closed ideal of L1() whose hull is Ω. Moreover, the -prime ideals of Ll() coincide with the kernels Ker Ω, where Ω denotes an arbitrary orbit (not necessarily closed) in *.

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39

Hartney, J. F. T. "On the decomposition of the s-radical of a near-ring." Proceedings of the Edinburgh Mathematical Society 33, no. 1 (February 1990): 11–22. http://dx.doi.org/10.1017/s0013091500028844.

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This paper concerns a Jacobson-type radical for the near-ring N. This radical, denoted by Js(N) has an external representation on a type-0N-group of a very special kind. Such N-groups are said to be of type-s. The main objective of this paper is to decompose Js(N) as a sum Js(N) = J1/2(N) + A + B for N satisfying the descending chain condition for N-subgroups. In this decomposition J1/2(N) is nilpotent and A is the unique minimal ideal modulo which Js(N) is nilpotent.

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40

Sheikh-Mohseni, S., and F. Saeedi. "Camina Lie algebras." Asian-European Journal of Mathematics 11, no. 05 (October 2018): 1850063. http://dx.doi.org/10.1142/s1793557118500638.

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Let [Formula: see text] be a Lie algebra and [Formula: see text] be a proper ideal of [Formula: see text]. Then [Formula: see text] is called a Camina pair if [Formula: see text] for all [Formula: see text]. Also, [Formula: see text] is called a Camina Lie algebra if [Formula: see text] is a Camina pair. In this paper, we give some properties of Camina Lie algebras. Moreover, we show that a nilpotent Camina Lie algebra of finite dimension over an algebraically closed field is nilpotent with nilindex at most [Formula: see text].

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41

LANSKI, CHARLES. "FINITE HIGHER COMMUTATORS IN ASSOCIATIVE RINGS." Bulletin of the Australian Mathematical Society 89, no. 3 (September 27, 2013): 503–9. http://dx.doi.org/10.1017/s0004972713000890.

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AbstractIf $T$ is any finite higher commutator in an associative ring $R$, for example, $T= [[R, R] , [R, R] ] $, and if $T$ has minimal cardinality so that the ideal generated by $T$ is infinite, then $T$ is in the centre of $R$ and ${T}^{2} = 0$. Also, if $T$ is any finite, higher commutator containing no nonzero nilpotent element then $T$ generates a finite ideal.

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42

Seddighin, Morteza. "On the ideals of extended quasi-nilpotent Banach algebras." International Journal of Mathematics and Mathematical Sciences 14, no. 3 (1991): 481–84. http://dx.doi.org/10.1155/s0161171291000662.

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Given a quasi-nilpotent Banach algebraA, we will use the results of Seddighin [2], to study the properties of elements which belong to a proper closed two sided ideal ofA¯andA¯¯. HereA¯is the extension ofAto a Banach Algebra with identity.

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43

Wang, Zhihua. "The corepresentation ring of a pointed Hopf algebra of rank one." Journal of Algebra and Its Applications 17, no. 12 (December 2018): 1850236. http://dx.doi.org/10.1142/s0219498818502365.

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Let [Formula: see text] be an arbitrary pointed Hopf algebra of rank one and [Formula: see text] the group of group-like elements of [Formula: see text]. In this paper, we give the decomposition of a tensor product of finite dimensional indecomposable right [Formula: see text]-comodules into a direct sum of indecomposables. This enables us to describe the corepresentation ring of [Formula: see text] in terms of generators and relations. Such a ring is not commutative if [Formula: see text] is not abelian. We describe all nilpotent elements of the corepresentation ring of [Formula: see text] if [Formula: see text] is a finite abelian group or a particular Hamiltonian group. In this case, all nilpotent elements of the corepresentation ring form a principal ideal which is either zero or generated by a nilpotent element of degree 2.

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44

Riley, D. M., and A. Shalev. "Restricted Lie Algebras and Their Envelopes." Canadian Journal of Mathematics 47, no. 1 (February 1, 1995): 146–64. http://dx.doi.org/10.4153/cjm-1995-008-7.

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AbstractLet L be a restricted Lie algebra over a field of characteristic p. Denote by u(L) its restricted enveloping algebra and by ωu(L) the augmentation ideal of u(L). We give an explicit description for the dimension subalgebras of L, namely those ideals of L defined by Dn(L) - L∩ωu(L)n for each n ≥ 1. Using this expression we describe the nilpotence index of ωU(L). We also give a precise characterisation of those L for which ωu(L) is a residually nilpotent ideal. In this case we show that the minimal number of elements required to generate an arbitrary ideal of u(L) is finitely bounded if and only if L contains a 1-generated restricted subalgebra of finite codimension. Subsequently we examine certain analogous aspects of the Lie structure of u(L). In particular we characterise L for which u(L) is residually nilpotent when considered as a Lie algebra, and give a formula for the Lie nilpotence index of u(L). This formula is then used to describe the nilpotence class of the group of units of u(L).

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45

Makarenko, N. Yu. "A Nilpotent Ideal in the Lie Rings with Automorphism of Prime Order." Siberian Mathematical Journal 46, no. 6 (November 2005): 1097–107. http://dx.doi.org/10.1007/s11202-005-0104-0.

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46

Kaniuth, Eberhard. "Induced characters, Mackey analysis and primitive ideal spaces of nilpotent discrete groups." Journal of Functional Analysis 240, no. 2 (November 2006): 349–72. http://dx.doi.org/10.1016/j.jfa.2006.04.019.

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47

ZANARDO, PAOLO. "ALMOST PERFECT LOCAL DOMAINS AND THEIR DOMINATING ARCHIMEDEAN VALUATION DOMAINS." Journal of Algebra and Its Applications 01, no. 04 (December 2002): 451–67. http://dx.doi.org/10.1142/s0219498802000288.

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A commutative ring R is said to be almost perfect if R/I is perfect for every nonzero ideal I of R. We prove that an almost perfect local domain R is dominated by a unique archimedean valuation domain V of its field of quotients Q if and only if the integral closure of R contains an ideal of V. We show how to construct almost perfect local domains dominated by finitely many archimedean valuation domains. We provide several examples illustrating various possible situations. In particular, we construct an almost perfect local domain whose maximal ideal is not almost nilpotent.

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48

Salemkar, Ali Reza, and Somaieh Alizadeh Niri. "BOUNDS FOR THE DIMENSION OF THE SCHUR MULTIPLIER OF A PAIR OF NILPOTENT LIE ALGEBRAS." Asian-European Journal of Mathematics 05, no. 04 (December 2012): 1250059. http://dx.doi.org/10.1142/s1793557112500593.

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Let (L, N) be a pair of finite-dimensional nilpotent Lie algebras, in which N is an ideal in L. In this paper we derive some inequalities for the dimension of the Schur multiplier of the pair (L, N) in terms of the dimension of the commutator subalgebra [L, N].

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Habibi, Mohammad, and Ahmad Moussavi. "Special properties of a skew triangular matrix ring with constant diagonal." Asian-European Journal of Mathematics 08, no. 03 (September 2015): 1550021. http://dx.doi.org/10.1142/s1793557115500217.

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In this paper, we study various annihilator properties in a ring [Formula: see text] with an endomorphism [Formula: see text] and some subrings of the skew triangular matrix rings [Formula: see text]. They allow the construction of rings with a nonzero nilpotent ideal of arbitrary index of nilpotency which have various zero-divisor properties.

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50

URSUL, MIHAIL, and MARTIN JURAS. "Notes on topological rings." Carpathian Journal of Mathematics 29, no. 2 (2013): 267–73. http://dx.doi.org/10.37193/cjm.2013.02.02.

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We prove that every infinite nilpotent ring R admits a ring topology T for which (R, T ) has an open totally bounded countable subring with trivial multiplication. A new example of a compact ring R for which R2 is not closed, is given. We prove that every compact Bezout domain is a principal ideal domain.

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