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Journal articles on the topic 'Radicals of ideals'

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Published: 10 December 2022

Last updated: 29 January 2023

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1

Sands, A. D. "Radicals and one-sided ideals." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 103, no. 3-4 (1986): 241–51. http://dx.doi.org/10.1017/s0308210500018898.

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The correspondence between radicals of associative rings and A-radicals is studied. It is shown that corresponding to each A-radical there is an interval of radicals and that each radical belongs to exactly one such interval. The question of the nature of the radical of a one-sided ideal is considered. It is shown that the radicals such that the radical of a one-sided ideal is always a one-sided ideal are those which contain their associated A-radicals. Radicals such that the radical of a one-sided ideal always equals the intersection of a left ideal and a right ideal are described, as are those A-radicals such that every associated radical has this property.

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2

Kumbhojkar, H. V. "Proper Fuzzification of Prime Ideals of a Hemiring." Advances in Fuzzy Systems 2012 (2012): 1–8. http://dx.doi.org/10.1155/2012/801650.

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Prime fuzzy ideals, prime fuzzyk-ideals, and prime fuzzyh-ideals are roped in one condition. It is shown that this way better fuzzification is achieved. Other major results of the paper are: every fuzzy ideal (resp.,k-ideal,h-ideal) is contained in a prime fuzzy ideal (resp.,k-ideal,h-ideal). Prime radicals and nil radicals of a fuzzy ideal are defined; their relationship is established. The nil radical of a fuzzyk-ideal (resp., anh-ideal) is proved to be a fuzzyk-ideal (resp.,h-ideal). The correspondence theorems for different types of fuzzy ideals of hemirings are established. The concept of primary fuzzy ideal is introduced. Minimum imperative for proper fuzzification is suggested and it is shown that the fuzzifications introduced in this paper are proper fuzzifications.

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3

Bell, Allen D., Shubhangi S. Stalder, and Mark L. Teply. "Prime ideals and radicals in semigroup-graded rings." Proceedings of the Edinburgh Mathematical Society 39, no. 1 (February 1996): 1–25. http://dx.doi.org/10.1017/s0013091500022720.

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In this paper we study the ideal structure of the direct limit and direct sum (with a special multiplication) of a directed system of rings; this enables us to give descriptions of the prime ideals and radicals of semigroup rings and semigroup-graded rings.We show that the ideals in the direct limit correspond to certain families of ideals from the original rings, with prime ideals corresponding to “prime” families. We then assume the indexing set is a semigroup ft with preorder defined by α≺β if β is in the ideal generated by α, and we use the direct sum to construct an Ω-graded ring; this construction generalizes the concept of a strong supplementary semilattice sum of rings. We show the prime ideals in this direct sum correspond to prime ideals in the direct limits taken over complements of prime ideals in Ω when two conditions are satisfied; one consequence is that when these conditions are satisfied, the prime ideals in the semigroup ring S[ft] correspond bijectively to pairs (Φ, Q) with Φ a prime ideal of Ω and Q a prime ideal of S. The two conditions are satisfied in many bands and in any commutative semigroup in which the powers of every element become stationary. However, we show that the above correspondence fails when Ω is an infinite free band, by showing that S[Ω] is prime whenever S is.When Ω satisfies the above-mentioned conditions, or is an arbitrary band, we give a description of the radical of the direct sum of a system in terms of the radicals of the component rings for a class of radicals which includes the Jacobson radical and the upper nil radical. We do this by relating the semigroup-graded direct sum to a direct sum indexed by the largest semilattice quotient of Ω, and also to the direct product of the component rings.

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4

MESYAN, ZACHARY. "THE IDEALS OF AN IDEAL EXTENSION." Journal of Algebra and Its Applications 09, no. 03 (June 2010): 407–31. http://dx.doi.org/10.1142/s0219498810003999.

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Given two unital associative rings R ⊆ S, the ring S is said to be an ideal (or Dorroh) extension of R if S = R ⊕ I, for some ideal I ⊆ S. In this note, we investigate the ideal structure of an arbitrary ideal extension of an arbitrary ring R. In particular, we describe the Jacobson and upper nil radicals of such a ring, in terms of the Jacobson and upper nil radicals of R, and we determine when such a ring is prime and when it is semiprime. We also classify all the prime and maximal ideals of an ideal extension S of R, under certain assumptions on the ideal I. These are generalizations of earlier results in the literature.

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5

Sands, A. D. "Radicals and one-sided ideals: an addendum." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 108, no. 1-2 (1988): 89–90. http://dx.doi.org/10.1017/s0308210500026548.

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SynopsisIn the paper referred to in the title [2] an open question was raised concerning the equality of the largest left hereditary radical and the largest right hereditary radical contained in each of certain radicals. In this addendum an affirmative answer is provided to this question.

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6

Wang, Zhudeng, Fengming Dai, and Yandong Yu. "Radicals of TL-ideals." Fuzzy Sets and Systems 121, no. 2 (July 2001): 301–14. http://dx.doi.org/10.1016/s0165-0114(99)00147-5.

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7

Becker, Eberhard, Rudolf Grobe, and Michael Niermann. "Radicals of binomial ideals." Journal of Pure and Applied Algebra 117-118 (May 1997): 41–79. http://dx.doi.org/10.1016/s0022-4049(97)00004-2.

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8

Malik, D. S., and John N. Mordeson. "Radicals of fuzzy ideals." Information Sciences 65, no. 3 (November 1992): 239–52. http://dx.doi.org/10.1016/0020-0255(92)90122-o.

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9

Ferrero, Miguel, and Edmund R. Puczyłowski. "The singular ideal and radicals." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 64, no. 2 (April 1998): 195–209. http://dx.doi.org/10.1017/s1446788700001695.

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AbstractSome properties of the singular ideal are established. In particular its behaviour when passing to one-sided ideals is studied. Obtained results are applied to study some radicals related to the singular ideal. In particular a radical S such that for every ring R, S(R) and R/S(R) are close to being a singular ring and a non-singular ring, respectively, is constructed.

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10

Gu, Ze. "On f - prime radical in ordered semigroups." Open Mathematics 16, no. 1 (June 7, 2018): 574–80. http://dx.doi.org/10.1515/math-2018-0053.

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AbstractIn this paper, we introduce the concepts of f-prime ideals, f-semiprime ideals and f-prime radicals in ordered semigroups. Furthermore, some results on f-prime radicals and f-primary decomposition of an ideal in an ordered semigroup are obtained.

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11

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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12

Ravi, M. S. "Regularity of ideals and their radicals." Manuscripta Mathematica 68, no. 1 (December 1990): 77–87. http://dx.doi.org/10.1007/bf02568752.

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13

Hong, Chan Yong, Tai Keun Kwak, and S. Tariq Rizvi. "Rigid Ideals and Radicals of Ore Extensions." Algebra Colloquium 12, no. 03 (September 2005): 399–412. http://dx.doi.org/10.1142/s1005386705000374.

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For an endomorphism σ of a ring R, Krempa called σ a rigid endomorphism if aσ(a)=0 implies a=0 for a∈R. A ring R is called rigid if there exists a rigid endomorphism of R. In this paper, we extend the σ-rigid property of a ring R to an ideal of R. For a σ-ideal I of a ring R, we call I a σ-rigid ideal if aσ(a)∈I implies a∈I for a∈R. We characterize σ-rigid ideals and study related properties. The connections of the prime radical and the upper nil radical of R with the prime radical and the upper nil radical of the Ore extension R[x;σ,δ], respectively, are also investigated.

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14

Sunstein, Cass. "Which Radicals?" Michigan Law Review, no. 117.6 (2019): 1215. http://dx.doi.org/10.36644/mlr.117.6.which.

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15

Anderson, T., K. Kaarli, and R. Wiegandt. "On left strong radicals of near-rings." Proceedings of the Edinburgh Mathematical Society 31, no. 3 (October 1988): 447–56. http://dx.doi.org/10.1017/s0013091500037639.

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In this paper we shall deal with radicals γ of near-rings such that for every near-ring N its radical γ(N) contains all left ideals (left invariant subgroups, respectively) I of N with I∈γ. At first, examples of such radicals will be given. Then we shall prove that these radicals are hypersolvable and have hereditary semisimple classes.

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16

Yua, Yandong, and Zhudeng Wang. "TL-subrings and TL-ideals, Part 4: Radicals of TL-ideals." Fuzzy Sets and Systems 88, no. 2 (June 1997): 227–36. http://dx.doi.org/10.1016/s0165-0114(96)00003-6.

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17

Laksov, D., and M. Rosenlund. "Radicals of ideals that are not the intersection of radical primes." Fundamenta Mathematicae 185, no. 1 (2005): 83–96. http://dx.doi.org/10.4064/fm185-1-6.

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18

AYGUN, Emin, and Betül ERDAL. "Radicals of soft intersectıonal ideals in semigroups." Cumhuriyet Science Journal 42, no. 1 (March 29, 2021): 115–22. http://dx.doi.org/10.17776/csj.738926.

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19

Tchernev, Alexandre B. "Criteria for equality of radicals of ideals." Journal of Algebra 267, no. 2 (September 2003): 396–403. http://dx.doi.org/10.1016/s0021-8693(03)00426-5.

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20

Kumar, Rajesh. "Fuzzy nil radicals and fuzzy primary ideals." Fuzzy Sets and Systems 43, no. 1 (September 1991): 81–93. http://dx.doi.org/10.1016/0165-0114(91)90023-j.

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21

Street, Ross. "Ideals, radicals, and structure of additive categories." Applied Categorical Structures 3, no. 2 (1995): 139–49. http://dx.doi.org/10.1007/bf00877633.

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22

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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23

Whelan, E. A. "Normalising elements and radicals, I." Bulletin of the Australian Mathematical Society 39, no. 1 (February 1989): 81–106. http://dx.doi.org/10.1017/s0004972700028008.

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In this paper we study rings and bimodules with no known one-sided chain conditions, but whose (two-sided) ideals and subbimodules are ‘nicely’ generated. We define bi-noetherian polycentral (BPC) and bi-noetherian polynormal (BPN) rings and bimodules, large classes of (almost always) non-noetherian objects, and put on record the basic facts about them. Any BPC ring is a BPN ring. In the case of rings we reduce their properties to properties of the prime ideals, and study the d.c.c. on (two-sided) ideals. We define both the artinian and bi-artinian radicals of a BPN ring, and use them to show that for BPN rings the intersections of the powers of both the Brown-McCoy and the Jacobson radicals are zero.

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24

Satvong, Narakon, Panuwat Luangchaisri, and Thawhat Changphas. "Radicals of generalized prime ideals in ternary semigroups." Discussiones Mathematicae - General Algebra and Applications 42, no. 2 (2022): 395. http://dx.doi.org/10.7151/dmgaa.1398.

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25

Neuhaus, Rolf. "Computation of real radicals of polynomial ideals — II." Journal of Pure and Applied Algebra 124, no. 1-3 (February 1998): 261–80. http://dx.doi.org/10.1016/s0022-4049(96)00103-x.

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26

Guangxing, Zeng. "Computation of generalized real radicals of polynomial ideals." Science in China Series A: Mathematics 42, no. 3 (March 1999): 272–80. http://dx.doi.org/10.1007/bf02879061.

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27

Nochefranca, L., and K. P. Shum. "Pseudo-symmetric ideals of semigroup and their radicals." Czechoslovak Mathematical Journal 48, no. 4 (December 1998): 727–35. http://dx.doi.org/10.1023/a:1022439706828.

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28

Birkenmeier, Gary, Henry Heatherly, and Enoch Lee. "Prime ideals and prime radicals in near-rings." Monatshefte f�r Mathematik 117, no. 3-4 (September 1994): 179–97. http://dx.doi.org/10.1007/bf01299701.

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29

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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30

Maxson, C. J., and L. Van Wyk. "The lattice of ideals of MR(R2)Ra commutative pir." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 52, no. 3 (June 1992): 368–82. http://dx.doi.org/10.1017/s1446788700035096.

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AbstractIn this paper we characterize the ideals of the centralizer near-ring N = MR(R2), where R is a commutative principle ideal ring. The characterization is used to determine the radicals Jυ(N) and the quotient structures N/ Jv(N), v = 0, 1, 2.

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31

Jun, Young Bae, Mehmet Ali Öztürk, and Chul Hwan Park. "Intuitionistic nil radicals of intuitionistic fuzzy ideals and Euclidean intuitionistic fuzzy ideals in rings." Information Sciences 177, no. 21 (November 2007): 4662–77. http://dx.doi.org/10.1016/j.ins.2007.03.020.

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32

LEE, ENOCH K. S. "PRIME IDEALS IN STRUCTURAL MATRIX NEAR-RINGS." Tamkang Journal of Mathematics 26, no. 1 (March 1, 1995): 31–40. http://dx.doi.org/10.5556/j.tkjm.26.1995.4376.

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This paper studies different types of prime ideals and their radicals in structural matrix near-rings. Relationships between various types of prime ideals of a near-ring and the corresponding structural matrix near-ring are given.

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33

Meyer, J. H. "Two-sided ideals in group near-rings." Journal of the Australian Mathematical Society 77, no. 3 (December 2004): 321–34. http://dx.doi.org/10.1017/s1446788700014464.

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AbstractThe two-sided ideals of group near-rings are characterized and studied. Various examples are presented to illustrate the interplay between ideals in the base near-ring R and the corresponding group near-ring R[G]. Some results concerning the Jacobson radicals of R[G] are also discussed.

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34

Wang, Dongming. "Unmixed and prime decomposition of radicals of polynomial ideals." ACM SIGSAM Bulletin 32, no. 4 (December 1998): 2–9. http://dx.doi.org/10.1145/307733.307734.

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35

Shavgulidze, N. E. "Radicals of l-rings and one-sided l-ideals." Journal of Mathematical Sciences 166, no. 5 (April 13, 2010): 682–90. http://dx.doi.org/10.1007/s10958-010-9884-2.

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36

Kumar, Rajesh. "Fuzzy nil radicals and fuzzy semiprimary ideals of rings." Fuzzy Sets and Systems 51, no. 3 (November 1992): 345–50. http://dx.doi.org/10.1016/0165-0114(92)90025-y.

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37

Kumbhojkar, H. V., and M. S. Bapat. "On prime and primary fuzzy ideals and their radicals." Fuzzy Sets and Systems 53, no. 2 (January 1993): 203–16. http://dx.doi.org/10.1016/0165-0114(93)90174-g.

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38

Katsabekis, Anargyros, Marcel Morales, and Apostolos Thoma. "Stanley–Reisner rings and the radicals of lattice ideals." Journal of Pure and Applied Algebra 204, no. 3 (March 2006): 584–601. http://dx.doi.org/10.1016/j.jpaa.2005.06.005.

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39

Xie, Xiang-Yun, and Jian Tang. "Fuzzy radicals and prime fuzzy ideals of ordered semigroups." Information Sciences 178, no. 22 (November 2008): 4357–74. http://dx.doi.org/10.1016/j.ins.2008.07.006.

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40

Boulier, François, Daniel Lazard, François Ollivier, and Michel Petitot. "Computing representations for radicals of finitely generated differential ideals." Applicable Algebra in Engineering, Communication and Computing 20, no. 1 (March 7, 2009): 73–121. http://dx.doi.org/10.1007/s00200-009-0091-7.

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41

KOSLER, KARL A. "EXTENDING TORSION RADICALS." Journal of Algebra and Its Applications 07, no. 01 (February 2008): 91–108. http://dx.doi.org/10.1142/s0219498808002680.

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Over an arbitrary ring R, a symmetric radical is shown to be strongly normalizing. For a fully semiprimary Noetherian ring R, a symmetric radical is normalizing if and only if the class of torsion factor rings of R is closed under ring isomorphisms. In case S is a strongly normalizing or normalizing extension ring of R, a symmetric radical for S is constructed as an extension of a symmetric radical for R. Applications address questions concerning the behavior of Krull dimension and linked prime ideals of S.

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42

FRANCE-JACKSON, HALINA. "ON SUPERNILPOTENT NONSPECIAL RADICALS." Bulletin of the Australian Mathematical Society 78, no. 1 (August 2008): 107–10. http://dx.doi.org/10.1017/s0004972708000518.

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AbstractLet ρ be a supernilpotent radical. Let ρ* be the class of all rings A such that either A is a simple ring in ρ or the factor ring A/I is in ρ for every nonzero ideal I of A and every minimal ideal M of A is in ρ. Let $\mathcal {L}\left ( \rho ^{\ast }\right ) $ be the lower radical determined by ρ* and let ρφ denote the upper radical determined by the class of all subdirectly irreducible rings with ρ-semisimple hearts. Le Roux and Heyman proved that $\mathcal {L}\left ( \rho ^{\ast }\right ) $ is a supernilpotent radical with $\rho \subseteq \mathcal {L}\left ( \rho ^{\ast }\right ) \subseteq \rho _{\varphi }$ and they asked whether $\mathcal {L} \left ( \rho ^{\ast }\right ) =\rho _{\varphi }$ if ρ is replaced by β, ℒ , 𝒩 or 𝒥 , where β, ℒ , 𝒩 and 𝒥 denote the Baer, the Levitzki, the Koethe and the Jacobson radical, respectively. In the present paper we will give a negative answer to this question by showing that if ρ is a supernilpotent radical whose semisimple class contains a nonzero nonsimple * -ring without minimal ideals, then $\mathcal {L}\left ( \rho ^{\ast }\right ) $ is a nonspecial radical and consequently $\mathcal {L}\left ( \rho ^{\ast }\right ) \neq \rho _{\varphi }$. We recall that a prime ring A is a * -ring if A/I is in β for every $0\neq I\vartriangleleft A$.

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43

Yeşіlot, Gürsel. "Radicals of submodules in modules." Studia Scientiarum Mathematicarum Hungarica 47, no. 4 (December 1, 2010): 445–47. http://dx.doi.org/10.1556/sscmath.2009.1149.

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The radical radN of a submodule N of a module M is defined as the intersection of all prime submodules containing N. In this paper, we show that if the ideals (N: M) and (L: M) are comaximal then the equality rad (N ∩ L) = radN ∩ rad L holds. We also discuss some properties of finitely generated R-modules.

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44

Lee, Enoch K. S. "On primeness and nilpotence in structural matrix near-rings." Proceedings of the Edinburgh Mathematical Society 39, no. 2 (June 1996): 345–56. http://dx.doi.org/10.1017/s0013091500023063.

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The structure of completely prime ideals in any structural matrix near-rings is determined. Partial descriptions are obtained for prime, nil, nilpotent, and locally nilpotent ideals of structural matrix near-rings. Their associated radicals are also studied in this paper.

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45

Esin, Songül, Müge Kanuni, Ayten Koç, Katherine Radler, and Kulumani M. Rangaswamy. "On Prüfer-like properties of Leavitt path algebras." Journal of Algebra and Its Applications 19, no. 07 (July 5, 2019): 2050122. http://dx.doi.org/10.1142/s0219498820501224.

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Prüfer domains and subclasses of integral domains such as Dedekind domains admit characterizations by means of the properties of their ideal lattices. Interestingly, a Leavitt path algebra [Formula: see text], in spite of being noncommutative and possessing plenty of zero divisors, seems to have its ideal lattices possess the characterizing properties of these special domains. In [The multiplicative ideal theory of Leavitt path algebras, J. Algebra 487 (2017) 173–199], it was shown that the ideals of [Formula: see text] satisfy the distributive law, a property of Prüfer domains and that [Formula: see text] is a multiplication ring, a property of Dedekind domains. In this paper, we first show that [Formula: see text] satisfies two more characterizing properties of Prüfer domains which are the ideal versions of two theorems in Elementary Number Theory, namely, for positive integers [Formula: see text], [Formula: see text] and [Formula: see text]. We also show that [Formula: see text] satisfies a characterizing property of almost Dedekind domains in terms of the ideals whose radicals are prime ideals. Finally, we give necessary and sufficient conditions under which [Formula: see text] satisfies another important characterizing property of almost Dedekind domains, namely, the cancellative property of its nonzero ideals.

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46

Bell, Allen D. "Prime ideals and radicals in rings graded by clifford semigroups." Communications in Algebra 25, no. 5 (January 1997): 1595–608. http://dx.doi.org/10.1080/00927879708825939.

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47

Hudson, T. "Radicals and prime ideals in limit subalgebras of AF algebras." Quarterly Journal of Mathematics 48, no. 190 (June 1, 1997): 213–33. http://dx.doi.org/10.1093/qjmath/48.190.213.

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48

Ferrero, Miguel, and Edmund R. Puczyłowski. "Prime ideals and radicals of centred extensions and tensor products." Israel Journal of Mathematics 94, no. 1 (January 1996): 381–401. http://dx.doi.org/10.1007/bf02762713.

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49

HUDSON, TIMOTHY D. "RADICALS AND PRIME IDEALS IN LIMIT SUBALGEBRAS OF AF ALGEBRAS." Quarterly Journal of Mathematics 48, no. 2 (1997): 213–33. http://dx.doi.org/10.1093/qmath/48.2.213.

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50

Wyk, L. Van. "Prime and special ideals in structural matrix rings over a ring without unity." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 45, no. 2 (October 1988): 220–26. http://dx.doi.org/10.1017/s1446788700030135.

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AbstractA. D. Sands showed that there is a 1–1 correspondence between the prime ideals of an arbitraty associative ring R and the complete matrix ring Mn(R) via P→ Mn(P). A structural matrix ring M(B, R) is the ring of all n × n matrices over R with 0 in the positions where the n × n boolean matrix B, B a quasi-order, has 0. The author characterized the special ideals of M(B, R′), in case R′ has unity, for certain special lasses of rings. In this note results of sands and the author are generalized to structural matrix rings over rings without unity. I t turns out that, although the class of prime simple rings is not a special class, Nagata's M-radical has the same form in structural matrix rings as the special radicals studied by the author.

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