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Journal articles on the topic 'Jacobson radical'

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Published: 4 June 2021

Last updated: 25 April 2022

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1

KOSLER, KARL A. "ON SYMMETRIC RADICALS OVER FULLY SEMIPRIMARY NOETHERIAN RINGS." Journal of Algebra and Its Applications 02, no. 03 (September 2003): 351–64. http://dx.doi.org/10.1142/s021949880300057x.

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Symmetric radicals over a fully semiprimary Noetherian ring R are characterized in terms of stability on bimodules and link closure of special classes of prime ideals. The notion of subdirect irreduciblity with respect to a torsion radical is introduced and is shown to be invariant under internal bonds between prime ideals. An analog of the Jacobson radical is produced which is properly larger than the Jacobson radical, yet satisfies the conclusion of Jacobson's conjecture for right fully semiprimary Noetherian rings.

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2

Ilić-Georgijević, Emil. "On graded special radicals of graded rings." Journal of Algebra and Its Applications 17, no. 06 (May 23, 2018): 1850109. http://dx.doi.org/10.1142/s0219498818501098.

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In this paper, a graded ring is a ring which is the direct sum of a family of its additive subgroups indexed by a nonempty set under the assumption that the product of homogeneous elements is again homogeneous. We study graded special radicals and special radicals of graded rings, but which contain the corresponding Jacobson radicals. There are two versions of this graded radical, which we name the graded over-Jacobson and the large graded over-Jacobson radical. We establish several characterizations of the graded over-Jacobson radical of a graded ring and also prove that the largest homogeneous ideal contained in the corresponding classical radical of a graded ring coincides with the large graded over-Jacobson radical of that ring.

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3

SMOKTUNOWICZ, AGATA. "A NOTE ON NIL AND JACOBSON RADICALS IN GRADED RINGS." Journal of Algebra and Its Applications 13, no. 04 (January 9, 2014): 1350121. http://dx.doi.org/10.1142/s0219498813501211.

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It was shown by Bergman that the Jacobson radical of a Z-graded ring is homogeneous. This paper shows that the analogous result holds for nil radicals, namely, that the nil radical of a Z-graded ring is homogeneous. It is obvious that a subring of a nil ring is nil, but generally a subring of a Jacobson radical ring need not be a Jacobson radical ring. In this paper, it is shown that every subring which is generated by homogeneous elements in a graded Jacobson radical ring is always a Jacobson radical ring. It is also observed that a ring whose all subrings are Jacobson radical rings is nil. Some new results on graded-nil rings are also obtained.

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4

Godloza, L., N. J. Groenewald, and W. A. Olivier. "On Jacobson Near-rings and Special Radicals." Algebra Colloquium 14, no. 01 (March 2007): 1–14. http://dx.doi.org/10.1142/s1005386707000028.

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In this paper, we construct special radicals using class pairs of near-rings. We establish necessary conditions for a class pair to be a special radical class. We then define Jacobson-type near-rings and show that in most cases the class of all near-rings of this type is a special radical class. Subsequently, we investigate the relationship between each Jacobson-type near-ring and the corresponding matrix near-ring.

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5

KELAREV, A. V. "ON THE STRUCTURE OF INCIDENCE RINGS OF GROUP AUTOMATA." International Journal of Algebra and Computation 14, no. 04 (August 2004): 505–11. http://dx.doi.org/10.1142/s0218196704001888.

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The Jacobson radical is one of the major tools used in the investigation of the structure of rings and ring constructions. Our main theorem gives a complete description of the Jacobson radicals of incidence rings of group automata for all finite nilpotent groups.

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6

Rao, Ravi Srinivasa, K. Siva Prasad, and T. Srinivas. "Kurosh-Amitsur Right Jacobson Radical of Type 0 for Right Near-Rings." International Journal of Mathematics and Mathematical Sciences 2008 (2008): 1–6. http://dx.doi.org/10.1155/2008/741609.

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By a near-ring we mean a right near-ring.J0r, the right Jacobson radical of type 0, was introduced for near-rings by the first and second authors. In this paper properties of the radicalJ0rare studied. It is shown thatJ0ris a Kurosh-Amitsur radical (KA-radical) in the variety of all near-ringsR, in which the constant partRcofRis an ideal ofR. So unlike the left Jacobson radicals of types 0 and 1 of near-rings,J0ris a KA-radical in the class of all zero-symmetric near-rings.J0ris nots-hereditary and hence not an ideal-hereditary radical in the class of all zero-symmetric near-rings.

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7

Munn, W. D. "The Jacobson radical of a band ring." Mathematical Proceedings of the Cambridge Philosophical Society 105, no. 2 (March 1989): 277–83. http://dx.doi.org/10.1017/s0305004100067761.

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A band is a semigroup in which every element is idempotent. In this note we give an explicit description of the Jacobson radical of the semigroup ring of a band over a ring with unity. It is shown, further, that this radical is nil if and only if the Jacobson radical of the coefficient ring is nil. For the particular case of a normal band (see below for the definition) the Jacobson radical of the band ring is nilpotent if and only if the Jacobson radical of the coefficient ring is nilpotent; but this result does not extend to arbitrary bands.

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8

Behr, Erazm J. "Jacobson radical of filtered algebras." Proceedings of the American Mathematical Society 98, no. 4 (April 1, 1986): 545. http://dx.doi.org/10.1090/s0002-9939-1986-0861746-7.

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9

Najaryan, N. G. "HOMOGENEOUS IDEALS AND JACOBSON RADICAL." Proceedings of the YSU A: Physical and Mathematical Sciences 51, no. 2 (243) (June 15, 2017): 193–95. http://dx.doi.org/10.46991/pysu:a/2017.51.2.193.

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In this paper the Jacobson radical of an algebra$F\langle X\rangle / H$ is studied, where FhXi is a free associative algebra of countable rank over infinite field $F$ and $H$ is a homogeneous ideal of the algebr$F\langle X\rangle$. The following theorem is proved: the Jacobson radical of an algebra $F\langle X\rangle / H$ is a nil ideal.

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10

Olson, D. M., and R. Lidl. "A uniformly strongly prime radical." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 43, no. 1 (August 1987): 95–102. http://dx.doi.org/10.1017/s1446788700029013.

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AbstractThe class of all uniformly strongly prime rings is shown to be a special class of rings which generates a radical class which properly contains both the right and left strongly prime radicals and which is independent of the Jacobson and Brown-McCoy radicals.

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11

Abel, Mart, Mati Abel, and Paul Tammo. "Coincidence of topological Jacobson radicals in topological algebras." Acta et Commentationes Universitatis Tartuensis de Mathematica 21, no. 2 (December 22, 2017): 239–47. http://dx.doi.org/10.12697/acutm.2017.21.16.

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12

Kelarev, A. V. "On the Jacobson radical of semigroup rings of commutative semigroups." Mathematical Proceedings of the Cambridge Philosophical Society 108, no. 3 (November 1990): 429–33. http://dx.doi.org/10.1017/s0305004100069322.

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Many authors have considered the radicals of semigroup rings of commutative semigroups. A list of the papers pertaining to this field is contained in [4]. In [1] Amitsur proved that, for any associative ring R and for every free commutative semigroup S, the equalities B(RS) = B(R)S and L(RS) = L(R)S hold, where B is the Baer radical and L is the Levitsky radical. A natural problem which arises is to describe semigroup rings RS such that π(RS) = π(R)S, where π is one of the most important radicals. For the Baer and Levitsky radicals and commutative semigroups a complete solution of the above problem follows from theorems 2·8 and 3·1 of [15].

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13

Okniński, J., and P. Wauters. "Radicals of semigroup rings of commutative semigroups." Mathematical Proceedings of the Cambridge Philosophical Society 99, no. 3 (May 1986): 435–45. http://dx.doi.org/10.1017/s0305004100064380.

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In this paper we determine radicals of semigroup rings R[S] where R is an associative, not necessarily commutative, ring and S is a commutative semigroup. We will restrict ourselves to the prime radical P, the Levitzki radical L and the Jacobson radical J. At the end we will also give a few comments on the Brown-McCoy radical U.

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14

Kelarev, A. V. "Hereditary radicals and bands of associative rings." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 51, no. 1 (August 1991): 62–72. http://dx.doi.org/10.1017/s1446788700033309.

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AbstractBands of associative rings were introduced in 1973 by Weissglass. For the radicals playing most essential roles in the structure theory (in particular, for those of Jacobson, Baer, Levitsky, Koethe) it is shown how to find the radical of a band of rings. The technique of the general Kurosh-Amitsur radical theory is used to consider many radicals simultaneously.

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15

Rao, Ravi Srinivasa, and K. Siva Prasad. "A radical for right near-rings: The right Jacobson radical of type-0." International Journal of Mathematics and Mathematical Sciences 2006 (2006): 1–13. http://dx.doi.org/10.1155/ijmms/2006/68595.

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The notions of a right quasiregular element and right modular right ideal in a near-ring are initiated. Based on theseJ0r(R), the right Jacobson radical of type-0 of a near-ringRis introduced. It is obtained thatJ0ris a radical map andN(R)⊆J0r(R), whereN(R)is the nil radical of a near-ringR. Some characterizations ofJ0r(R) are given and its relation with some of the radicals is also discussed.

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16

Naghipour, A. R., and A. H. Yamini. "A GENERALIZATION OF THE JACOBSON RADICAL." Bulletin of the Korean Mathematical Society 41, no. 4 (November 1, 2004): 599–608. http://dx.doi.org/10.4134/bkms.2004.41.4.599.

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17

Puczylowski, E. R., and H. Zand. "On squares of Jacobson radical rings." Bulletin of the Australian Mathematical Society 53, no. 2 (April 1996): 299–303. http://dx.doi.org/10.1017/s0004972700017020.

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18

SMOKTUNOWICZ, AGATA, and ALEXANDER A. YOUNG. "JACOBSON RADICAL ALGEBRAS WITH QUADRATIC GROWTH." Glasgow Mathematical Journal 55, A (October 2013): 135–47. http://dx.doi.org/10.1017/s0017089513000554.

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AbstractWe show that over every countable algebraically closed field $\mathbb{K}$ there exists a finitely generated $\mathbb{K}$-algebra that is Jacobson radical, infinite-dimensional, generated by two elements, graded and has quadratic growth. We also propose a way of constructing examples of algebras with quadratic growth that satisfy special types of relations.

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19

Wallace, D. A. R. "THE JACOBSON RADICAL OF CLASSICAL RINGS." Bulletin of the London Mathematical Society 24, no. 6 (November 1992): 616–17. http://dx.doi.org/10.1112/blms/24.6.616.

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20

Rahman, Saifur, and Helen K. Saikia. "Fuzzy Small Submodule and Jacobson -Radical." International Journal of Mathematics and Mathematical Sciences 2011 (2011): 1–12. http://dx.doi.org/10.1155/2011/980320.

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Using the notion of fuzzy small submodules of a module, we introduce the concept of fuzzy coessential extension of a fuzzy submodule of a module. We attempt to investigate various properties of fuzzy small submodules of a module. A necessary and sufficient condition for fuzzy small submodules is established. We investigate the nature of fuzzy small submodules of a module under fuzzy direct sum. Fuzzy small submodules of a module are characterized in terms of fuzzy quotient modules. This characterization gives rise to some results on fuzzy coessential extensions. Finally, a relation between small -submodules and Jacobson -radical is established.

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21

CAI, CHUANREN, and GUANGQUAN GUO. "HOPF-JACOBSON RADICAL FOR COMODULE ALGEBRAS." Chinese Annals of Mathematics 20, no. 02 (April 1999): 261–70. http://dx.doi.org/10.1142/s0252959999000291.

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22

Yaqub, Adil. "Subperiodic rings with commutative Jacobson radical." International Journal of Algebra 8 (2014): 577–86. http://dx.doi.org/10.12988/ija.2014.4775.

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23

LAWRENCE, JOHN. "THE JACOBSON RADICAL OF TENSOR PRODUCTS." Quarterly Journal of Mathematics 42, no. 1 (1991): 203–8. http://dx.doi.org/10.1093/qmath/42.1.203.

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24

Chatters, A. W., and C. R. Hajarnavis. "Noetherian rings with projective jacobson radical." Communications in Algebra 13, no. 6 (January 1985): 1359–66. http://dx.doi.org/10.1080/00927878508823225.

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25

Lee, Yu Lee. "A note on the Jacobson radical." Proceedings of the American Mathematical Society 118, no. 2 (February 1, 1993): 337. http://dx.doi.org/10.1090/s0002-9939-1993-1131037-1.

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26

TUMURBAT, S. "ON SPECIAL RADICALS COINCIDING ON SIMPLE RINGS AND ON POLYNOMIAL RINGS." Journal of Algebra and Its Applications 02, no. 01 (March 2003): 51–56. http://dx.doi.org/10.1142/s0219498803000362.

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Answering a problem of M. Ferrero, we construct a special radical δ such that δ is contained in the Jacobson radical J, δ and J coincides on simple rings and on polynomial rings, but δ ≠ J. For two special radicals with the above conditions, we give a criterion of their coincidence.

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27

Zhu, Yongwen. "On the Jacobson Radical of an (m,n)-Semiring." Algebra 2013 (October 7, 2013): 1–9. http://dx.doi.org/10.1155/2013/272104.

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The notion of n-ary semimodules is introduced so that the Jacobson radical of an (m,n)-semiring is studied and some well-known results concerning the Jacobson radical of a ring (a semiring or a ternary semiring) are generalized to an (m,n)-semiring.

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28

Munn, W. D. "The algebra of a commutative semigroup over a commutative ring with unity." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 99, no. 3-4 (1985): 387–98. http://dx.doi.org/10.1017/s0308210500014396.

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SynopsisA new description is provided for the nil radical of the algebra RS of a commutative semigroup S over a commutative ring R with a 1. It is shown that the Jacobson radical of RS is nil if the Jacobson radical of R is nil and that the converse holds in the case where S is periodic.

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29

Arapović, Miroslav. "Approximation Theorems for Manis Valuations." Canadian Mathematical Bulletin 28, no. 2 (June 1, 1985): 184–89. http://dx.doi.org/10.4153/cmb-1985-019-5.

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AbstractThroughout this paper rings are understood to be commutative with unity. In this paper we prove the general approximation theorem for valuations whose infinite ideals have large Jacobson radicals. We give an example in which it is shown that approximation theorems for Manis valuations do not hold in the general case. Also we prove that every valuation pair (Rv, Pv) of a total quotient ring T(R) whose infinite ideal has large Jacobson radical is a Prüfer valuation pair.

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30

Velasco, M. Victoria. "The Jacobson radical of an evolution algebra." Journal of Spectral Theory 9, no. 2 (October 24, 2018): 601–34. http://dx.doi.org/10.4171/jst/257.

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31

Karpilovsky, G. "The Jacobson radical of monoid-graded algebras." Tsukuba Journal of Mathematics 16, no. 1 (June 1992): 19–52. http://dx.doi.org/10.21099/tkbjm/1496161828.

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32

Iovanov, Miodrag Cristian. "Semiartinian Profinite Algebras have Nilpotent Jacobson Radical." Algebras and Representation Theory 17, no. 4 (July 10, 2013): 1145–54. http://dx.doi.org/10.1007/s10468-013-9438-7.

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33

Prest, Mike, and Jan Schröer. "Serial functors, Jacobson radical and representation type." Journal of Pure and Applied Algebra 170, no. 2-3 (May 2002): 295–307. http://dx.doi.org/10.1016/s0022-4049(01)00133-5.

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34

Kelarev, A. V. "The Jacobson radical of commutative semigroup rings." Journal of Algebra 150, no. 2 (August 1992): 378–87. http://dx.doi.org/10.1016/s0021-8693(05)80037-7.

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35

Davidson, Kenneth R., and John Lindsay Orr. "The Jacobson radical of a CSL algebra." Transactions of the American Mathematical Society 344, no. 2 (February 1, 1994): 925–47. http://dx.doi.org/10.1090/s0002-9947-1994-1250816-9.

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36

KIM, HONG KEE, NAM KYUN KIM, and YANG LEE. "WEAKLY DUO RINGS WITH NIL JACOBSON RADICAL." Journal of the Korean Mathematical Society 42, no. 3 (May 1, 2005): 457–70. http://dx.doi.org/10.4134/jkms.2005.42.3.457.

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37

Donsig, Allan P., Aristides Katavolos, and Antonios Manoussos. "The Jacobson Radical for Analytic Crossed Products." Journal of Functional Analysis 187, no. 1 (December 2001): 129–45. http://dx.doi.org/10.1006/jfan.2001.3819.

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38

Jespers, E., A. V. Kelarev, and J. Okniński. "ON THE JACOBSON RADICAL OF GRADED RINGS." Communications in Algebra 29, no. 5 (April 30, 2001): 2185–91. http://dx.doi.org/10.1081/agb-100002177.

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39

Glavatsky, S. T., A. V. Mikhalev, and V. V. Tenzina. "The topological jacobson radical of rings. I." Journal of Mathematical Sciences 185, no. 2 (July 26, 2012): 207–20. http://dx.doi.org/10.1007/s10958-012-0910-4.

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40

Glavatsky, S. T., A. V. Mikhalev, and V. V. Tenzina. "The topological jacobson radical of rings. II." Journal of Mathematical Sciences 185, no. 3 (August 1, 2012): 381–88. http://dx.doi.org/10.1007/s10958-012-0922-0.

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41

Tuganbaev, D. A. "The Jacobson radical and Laurent series rings." Journal of Mathematical Sciences 152, no. 2 (July 2008): 304–6. http://dx.doi.org/10.1007/s10958-008-9051-1.

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42

Srivastava, J. B., and Anand Bala. "Group algebras primitive modulo the Jacobson radical." Indagationes Mathematicae (Proceedings) 89, no. 4 (1986): 479–85. http://dx.doi.org/10.1016/1385-7258(86)90032-6.

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43

Khemphet, Anchalee, and Justin R. Peters. "Semicrossed Products of the Disk Algebra and the Jacobson Radical." Canadian Mathematical Bulletin 57, no. 1 (March 14, 2014): 80–89. http://dx.doi.org/10.4153/cmb-2012-018-8.

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Abstract We consider semicrossed products of the disk algebra with respect to endomorphisms defined by finite Blaschke products. We characterize the Jacobson radical of these operator algebras. Furthermore, in the case that the finite Blaschke product is elliptic, we show that the semicrossed product contains no nonzero quasinilpotent elements. However, if the finite Blaschke product is hyperbolic or parabolic with positive hyperbolic step, the Jacobson radical is nonzero and a proper subset of the set of quasinilpotent elements.

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44

Pogorzały, Zygmunt. "Auslander–Reiten Orbit Algebras for Self-Injective Nakayama Algebras." Algebra Colloquium 12, no. 02 (June 2005): 351–60. http://dx.doi.org/10.1142/s1005386705000337.

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The main result of this paper is the following: Let A be a self-injective Nakayama K-algebra, which is basic and connected. Suppose that A is a right Ae-module of τAe-period 1. (1) If A is an algebra whose Jacobson radical square is zero, then [Formula: see text]. (2) If A is an algebra whose Jacobson radical square is not zero, then [Formula: see text] for some positive integer n.

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45

BIRKENMEIER, GARY F., HENRY E. HEATHERLY, and ENOCH K. S. LEE. "SPECIAL RADICALS FOR NEAR-RING." Tamkang Journal of Mathematics 27, no. 4 (October 12, 2020): 281–88. http://dx.doi.org/10.5556/j.tkjm.27.1996.3956.

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The concept of a special radical for near-rings has been treated in several nonequivalent, but related, ways in the recent literature. We use the version due to K. Kaarli to establish that various prime radicals and the nil radical are special radicals on the class A of all near-rings which satisfy an extended version of the Andrunakievich Lemma. Since A includes all d.g. near-rings—and much more—these results significantly extend results previously obtained by Kaarli and by Groenewald. We also obtain special radical results for the Jacobson type radicals 30 and 3 1 , albeit on less extensive classes. Examples are given which illustrate and delimit the theory developed.

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46

FRANCE-JACKSON, HALINA. "ON SUPERNILPOTENT RADICALS WITH THE AMITSUR PROPERTY." Bulletin of the Australian Mathematical Society 80, no. 3 (June 29, 2009): 423–29. http://dx.doi.org/10.1017/s0004972709000380.

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AbstractA radical α has the Amitsur property if α(A[x])=(α(A[x])∩A)[x] for all rings A. For rings R⊆S with the same unity, we call S a finite centralizing extension of R if there exist b1,b2,…,bt∈S such that S=b1R+b2R+⋯+btR and bir=rbi for all r∈R and i=1,2,…,t. A radical α is FCE-friendly if α(S)∩R⊆α(R) for any finite centralizing extension S of a ring R. We show that if α is a supernilpotent radical whose semisimple class contains the ring ℤ of all integers and α is FCE-friendly, then α has the Amitsur property. In this way the Amitsur property of many well-known radicals such as the prime radical, the Jacobson radical, the Brown–McCoy radical, the antisimple radical and the Behrens radical can be established. Moreover, applying this condition, we will show that the upper radical 𝒰(*k) generated by the essential cover *k of the class * of all *-rings has the Amitsur property and 𝒰(*k)(A[x])=𝒰(*k)(A)[x], where a semiprime ring R is called a *-ring if the factor ring R/I is prime radical for every nonzero ideal I of R. The importance of *-rings stems from the fact that a *-ring A is Jacobson semisimple if and only if A is a primitive ring.

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47

Sharma, Ram Parkash, Richa Sharma, and Madhu. "Radicals of semirings." Asian-European Journal of Mathematics 13, no. 07 (August 6, 2019): 2050138. http://dx.doi.org/10.1142/s1793557120501387.

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It is shown that the classes [Formula: see text] and [Formula: see text] of semirings are radical classes, where [Formula: see text] is the class of subtractive-simple right [Formula: see text]-semimodules and [Formula: see text] is the class of right [Formula: see text]-semimodules isomorphic to [Formula: see text] for some maximal-subtractive right ideal [Formula: see text] of [Formula: see text]. We define the lower Jacobson Bourne radical [Formula: see text] and upper Jacobson Bourne radical [Formula: see text] of [Formula: see text]. For a semiring [Formula: see text], [Formula: see text] holds, where [Formula: see text] is the Jacobson Bourne radical of [Formula: see text]. The radical [Formula: see text] and also coincides with [Formula: see text], if we restrict the class [Formula: see text] to additively cancellative semimodules[Formula: see text] The upper radical [Formula: see text] and [Formula: see text][Formula: see text], if [Formula: see text] is additively cancellative. Further, [Formula: see text], if [Formula: see text] is a commutative semiring with [Formula: see text] The subtractive-primitiveness and subtractive-semiprimitiveness of [Formula: see text] are closely related to the upper radical [Formula: see text] Finally, we show that [Formula: see text]-semisimplicity of semirings are Morita invariant property with some restrictions.

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48

Majumdar, Subrata, and Kalyan Kumar Dey. "Three Distinct Non-Hereditary Radicals Which Coincide with the Classical Radical for Rings with D.C.C." GANIT: Journal of Bangladesh Mathematical Society 35 (June 28, 2016): 1–5. http://dx.doi.org/10.3329/ganit.v35i0.28560.

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Majumdar and Paul [3] introduced and studied a new radical E defined as the upper radical determined by the class of all rings each of whose ideals is idempotent. In this paper the authors continue the study further and also study the join radical and the intersection radical (due to Leavitt) obtained from E and the Jacobson radical J. These have been denoted by E + J and EJ respectively. The radical and the semisimple rings corresponding to E + J and EJ have been obtained. Both of these radicals coincide with the classical nil radical for Artinian rings. Important properties of these radicals and their position among the well-known special radicals have been investigated. It has been proved that E, EJ and E + J are non-hereditary. It has also been proved as an independent result that the nil radical N is not dual, i.e., N ? N?.GANIT J. Bangladesh Math. Soc.Vol. 35 (2015) 1-11

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49

Ilić-Georgijević, Emil. "Notes on the graded Jacobson radical: a graded version of the Jacobson stable set." Communications in Algebra 48, no. 6 (February 9, 2020): 2624–31. http://dx.doi.org/10.1080/00927872.2020.1721730.

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50

Dăscălescu, S., and L. van Wyk. "A note on intermediate normalising extensions." Bulletin of the Australian Mathematical Society 50, no. 2 (October 1994): 313–16. http://dx.doi.org/10.1017/s0004972700013769.

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Abstract:

We prove that the following ring-theoretic properties are shared by the two rings involved in a normalising extension R ⊂ S, and that these properties are inherited by any intermediate extension: semilocal, left perfect, semiprimary. This transfer fails for the nilpotency of the Jacobson radical. However, if the normalising set is a basis for the left R-module S, then the nilpotency of the Jacobson radical behaves in the same way as the three properties mentioned above.

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