DOCSLIB.ORG

Quantum Groupoids Acting on Semiprime Algebras

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Hindawi Publishing Corporation Advances in Mathematical Physics Volume 2011, Article ID 546058, 9 pages doi:10.1155/2011/546058

Research Article

Quantum Groupoids Acting on Semiprime Algebras

Ineˆs Borges1 and Christian Lomp2

  • 1 Instituto Superior De Contabilidade e Administra c ¸ao de Coimbra, Quinta Agr ı cola-Bencanta,
  • ˜
  • ´

3040-316 Coimbra, Portugal

  • 2 Departamento de Matematica, Faculdade de Ciencias, Universidade do Porto, Rua Campo Alegre 687,
  • ´
  • ˆ

4169-007 Porto, Portugal

Correspondence should be addressed to Christian Lomp, [email protected] Received 30 March 2011; Accepted 5 June 2011 Academic Editor: Olaf Lechtenfeld Copyright q 2011 I. Borges and C. Lomp. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Following Linchenko and Montgomery’s arguments we show that the smash product of an involutive weak Hopf algebra and a semiprime module algebra, satisfying a polynomial identity, is semiprime.

1. Introduction

Group actions, Lie algebras acting as derivations and finite group gradings are typical examples of Hopf algebra actions which have been studied for many years. Several generalizations of Hopf algebras have emerged in recent years, like weak Hopf algebras ꢀor quantum

  • groupoidsꢁ introduced by Bohm et al. ꢂ1ꢃ. The action of such objects on algebras, as given by
  • ¨

quantum groupoids acting on C∗-algebras, ꢂ2ꢃ or weak Hopf algebras arising from Jones towers ꢂ3ꢃ are particularly interesting. New examples of weak Hopf algebras arose from double groupoids ꢂ4ꢃ, which were also used to find new weak Hopf actions ꢀsee ꢂ2ꢃꢁ.
A long-standing open problem in the theory of Hopf action is to show that the smash product A#H of a semiprime module algebra A and a semisimple Hopf algebra H is again semiprime ꢀsee ꢂ5ꢃꢁ ꢀan algebra A is semiprime if it does not contain nonzero nilpotent ideals.ꢁ. The case of A being commutative had been settled in ꢂ6ꢃ. The most recent partial answer to this problem has been given by Linchenko and Montgomery in ꢂ7ꢃ where they prove the semiprimness of A#H under the condition of A satisfying a polynomial identity. The purpose of this note is that their result carries over to actions of weak Hopf algebras. We reach more generality by considering actions of linear operators that satisfy certain intertwining relations with the regular multiplications on the algebra.

  • 2
  • Advances in Mathematical Physics

Let k be a commutative ring and let A be an associative unital k-algebra. For any

a ∈ A define two linear operators Lꢀaꢁ and Rꢀaꢁ in EndkꢀAꢁ given by ꢀLꢀaꢁ, xꢁ ꢄ ax and ꢀRꢀaꢁ, xꢁ ꢄ xa for all x ∈ A. We identify A with the subalgebra LꢀAꢁ of EndkꢀAꢁ generated

by all left multiplications Lꢀaꢁ and denote the subalgebra generated by all operators Lꢀaꢁ and Rꢀaꢁ by MꢀAꢁ, which is also sometimes referred to as the multiplication algebra of A. As a left LꢀAꢁ-module, A is isomorphic to LꢀAꢁ since we assume A to be unital. We will be interested in certain actions on an algebra A that may stem from a bialgebra or more generally a bialgebroid. The situation we will encounter is the one where we have an extension A ⊆ B

where B acts on A through a ring homomorphism φ : B → EndkꢀAꢁ such that ꢀaꢁφ ꢄ Lꢀaꢁ

for all a ∈ A. For the intrinsic properties of A under this action it is enough to look at the subalgebra φꢀBꢁ in EndkꢀAꢁ generated by this action and we might consider intermediate algebras LꢀAꢁ ⊆ B ⊆ EndkꢀAꢁ instead. Hence let B be a subalgebra of EndkꢀAꢁ that contains LꢀAꢁ. Then A becomes a cyclic faithful left B-module by evaluating endomorphisms, that is,

for all b ∈ B, a ∈ A : b·a :ꢄ ꢀb, aꢁ. Note that for any aꢂ ∈ A we have Lꢀaꢂꢁ·a ꢄ ꢀLꢀaꢂꢁ, aꢁ ꢄ aꢂa.

Since we assume A to be unital, the map Ψ : EndBꢀAꢁ → A with Ψꢀfꢁ ꢄ ꢀ1ꢁf, for all f ∈ EndBꢀAꢁ—evaluating endomorphisms at 1—is an injective ring homomorphism, since

for all f, g  ∈ EndBꢀAꢁ : Ψꢀf ◦ gꢁ ꢄ ꢀꢀ1ꢁfꢁg ꢄ ꢀꢀ1ꢁf · 1ꢁg ꢄ ꢀ1ꢁf · ꢀ1ꢁg ꢄ ΨꢀfꢁΨꢀgꢁ. Moreover if Ψꢀfꢁ ꢄ ꢀ1ꢁf ꢄ 0, then ꢀaꢁf ꢄ aꢀ1ꢁf ꢄ 0 and f ꢄ 0. The subalgebra ΨꢀEndBꢀAꢁꢁ can be described as the set of elements a ∈ A such that for any b ∈ B : b · a ꢄ ꢀb · 1ꢁa, which we will denote by AB. On one hand if a ꢄ ꢀ1ꢁf ꢄ Ψꢀfꢁ for some f ∈ EndBꢀAꢁ, then for any b ∈ B : b·a ꢄ ꢀb·1ꢁf ꢄ Lꢀꢀb·1ꢁꢁ·ꢀ1ꢁf ꢄ ꢀb·1ꢁa and on the other hand if a ∈ AB, then f ꢄ Rꢀaꢁ is left B-linear since for any b ∈ B and x ∈ A:

b · ꢀxꢁf ꢄ b · ꢀxaꢁ ꢄ b · ꢀLꢀxꢁ · aꢁ ꢄ ꢀbLꢀxꢁꢁ · a ꢄ ꢀbLꢀxꢁ · 1ꢁa

ꢀ1.1ꢁ

ꢄ ꢀb · ꢀLꢀxꢁ · 1ꢁꢁa ꢄ ꢀb · xꢁa ꢄ ꢀb · xꢁf.

Thus EndBꢀAꢁ ꢃ AB.

Let M be any left B-module and define MB ꢄ {m ∈ M : ∀b ∈ B : b · m ꢄ ꢀb · 1ꢁm}. With

the same argument as above one sees that ΨM : HomBꢀA, Mꢁ → MB with Ψꢀfꢁ ꢄ ꢀ1ꢁf is an isomorphism of abelian groups, hence yielding a left AB-module structure on MB. Moreover it is possible to show that HomBꢀA, −ꢁ is isomorphic to ꢀ−ꢁB as functors from B-Mod to AB- Mod.
A subset I of A is called B-stable if B · I ⊆ I. The B-stable left ideals are precisely the

ꢀleftꢁ B-submodules of A. In particular HomBꢀA, Iꢁ ꢃ IB ꢄ I ∩ AB, for any B-stable left ideal of A.

Examples 1.1.  The following list illustrates that our aproach reflects many interesting cases of algebras with actions.

ꢀ1ꢁ Let B ꢄ MꢀAꢁ be the multiplication algebra of A, then A is a faithful cyclic left
B-module. The left B-modules are precisely the A-bimodules, in particular the left

ideals of A are the two-sided ideals of A, and HomBꢀA, Mꢁ ꢃ MB ꢄ ZꢀMꢁ ꢄ {m ∈ M | ∀a ∈ A : am ꢄ ma} holds for any A-bimodule M. The operator algebra B is a quotient of the enveloping algebra Ae ꢄ A⊗Aop through the map a⊗b → Lꢀaꢁ◦Rꢀbꢁ, for all a ⊗ b ∈ Ae.

ꢀ2ꢁ Let G be a group acting as ꢀk-linearꢁ automorphisms on A, that is, there exists a

group homomorphism η : G → AutkꢀAꢁ. Set ga ꢄ ꢀηꢀgꢁ, aꢁ for any a ∈ A, g  ∈ G.

  • Advances in Mathematical Physics
  • 3

Define B ꢄ ꢀLꢀAꢁ ∪ {ηꢀgꢁ | g ∈ G}ꢁ ⊆ EndkꢀAꢁ. Then the left B-submodules I of A gx

  • are precisely the G-stable left ideals of A and HomBꢀA, Iꢁ ꢃ IB ꢄ {x ∈ I |
  • ꢄ x}.

B is a quotient of the skew group ring A#G whose underlying A-submodule is the free left A-module with basis {g | g ∈ G} and whose multiplication is

given by ꢀa#gꢁꢀb#hꢁ ꢄ agb#gh. Note that for any left A#G-module M we have, HomA#GꢀA, Mꢁ ꢄ HomBꢀA, Mꢁ ꢃ MG is the set of fixed elements of M.

ꢀ3ꢁ Let A be an k-algebra with involution ∗ and let B be the subalgebra of EndAꢀ ꢁ

generated by A and ∗. Since for any a ∈ A : Rꢀaꢁ ꢄ ∗ ◦ Lꢀa∗ꢁ ◦ ∗ we got MꢀAꢁ ⊆ B.

This means ꢀas it is well-knownꢁ that any left ideal of A which is stable under ∗ is a two-sided ideal. Note that B can be seen as the factor ring of the skew-group ring Ae#G where G ꢄ {id, ∗} is the cyclic group of order two and ∗ ∈ AutꢀAeꢁ is given

by ꢀa ⊗ bꢁ∗ :ꢄ b∗ ⊗ a∗.
ꢀ4ꢁ Let δ ∈ DerkꢀAꢁ be an k-linear derivation of A and consider B ꢄ ꢀLꢀAꢁ ∪ {δ}ꢁ ⊆
EndkꢀAꢁ. The left B-submodules of A are the left ideals I that satisfy δꢀIꢁ ⊆ I. The

operator algebra B is a factor of the ring of differential operator Aꢂz; δꢃ, which as

a left A-module is equal to Aꢂzꢃ and its multiplication is given by za ꢄ az δꢀaꢁ.

  • ꢀ
  • ꢀ

  • n
  • n

The map Aꢂz; δꢃ → B with iꢄ0 aizi → iꢄ0 Lꢀaiꢁ ◦ δi ∈ B is a surjective k-algebra homomorphism and for any left Aꢂz; δꢃ-module M we have HomAꢂz;δꢃꢀA, Mꢁ ꢄ HomBꢀA, Mꢁ ꢄ Mδ ꢄ {m ∈ M | zm ꢄ 0}. In particular EndAꢂz;δꢃꢀAꢁ ꢃ Aδ ꢄ Kerꢀδꢁ.

The subring Aδ of A is called the ring of constants of δ.
ꢀ5ꢁ Let H be an k-Hopf algebra action on A. Let us denote the action of an element

h ∈ H on A by λh ∈ EndkꢀAꢁ and define B ꢄ ꢀLꢀAꢁ ∪ {λh | h ∈ H}ꢁ ⊆ EndkꢀAꢁ.

The smash product A#H is an extension with additional module structure. Define

ϕ : A#H → EndkꢀAꢁ by ϕꢀa#hꢁ :ꢄ Lꢀaꢁ ◦ λh.

2. Linear Operators Acting on Algebras Satisfying a Polynomial Identity

Let LꢀAꢁ ⊆ B ⊆ EndkꢀAꢁ be any intermediate algebra as above.
The first technical lemma generalizes a corresponding result of Linchenko ꢂ8, Theorem 3.1ꢃ for Hopf actions and Nikshych ꢂ9, Theorem 6.1.3ꢃ for weak Hopf actions. Recall that an ideal whose elements are nilpotent is called a nil ideal.

Lemma 2.1. Let LꢀAꢁ ⊆ B ⊆ EndkꢀAꢁ and suppose that for all ψ ∈ B there exist m ≥ 1 and elements

ψ11, . . . , ψm1 , ψ12, . . . , ψm2 ∈ B such that

  • m
  • n

  • ꢅ
  • ꢅ

  • ꢁꢂ
  • ꢃꢄ

  • L
  • ψ, a

ꢄ

ψi1 ◦ Lꢀaꢁ ◦ ψi2, ψi2 ◦ ψi1 ∈ LꢀAꢁ

ꢀ2.1ꢁ

  • iꢄ1
  • iꢄ1

for any a ∈ A . If A is finite dimensional over a field of characteristic 0 and if I is a nil ideal, then B · I is nil. In particular the Jacobson radical of A is B -stable.

  • 4
  • Advances in Mathematical Physics

Proof. Denote the trace of a k-linear endomorphism f of A by Trꢀfꢁ. Let ψ ∈ B, a ∈ A. Using Trꢀf ghꢁ ꢄ Trꢀh f gꢁ and the hypotheses we get

ꢆꢆ

n

ꢅ

  • ꢁ ꢁꢂ
  • ꢃꢄꢄ
  • ꢁ ꢁ ꢄ
  • ꢄ
  • ꢁ ꢁ ꢄꢄ

Tr L ψ,   a

ꢄ Tr

ψi2 ◦ ψi1 ◦ Lꢀaꢁ ꢄ Tr L y  ◦ Lꢀaꢁ ꢄ Tr L ya

ꢀ2.2ꢁ

iꢄ1

for some y ∈ A. Suppose that a ∈ I with I a nil ideal, then ya ∈ I is nilpotent, hence

k

TrꢀLꢀꢀψ, aꢁꢁꢁ ꢄ TrꢀLꢀyaꢁꢁ ꢄ 0. For any k > 0 set zk :ꢄ ꢀψ, aꢁ . Then

n

  • ꢇ
  • ꢈ
  • ꢇ
  • ꢈ

ꢅ

  • ꢁꢂ
  • ꢃꢄ

zk ꢄ

  • L
  • ψ, a   , zk−1

ꢄ

ψi1, ai

ꢀ2.3ꢁ

iꢄ1

for ai ꢄ aꢀψi2, zk−1ꢁ. Since I is an ideal, ai ∈ I. Hence

n

  • ꢉ
  • ꢊ
  • ꢉ ꢉ ꢊꢊ
  • ꢉ ꢉꢇ
  • ꢈꢊꢊ

ꢅ

Tr Lꢀzꢁk ꢄ Tr L zk

ꢄ

  • Tr L ψi1, ai
  • ꢄ 0.

ꢀ2.4ꢁ

iꢄ1

Since A is finite dimensional, charꢀkꢁ ꢄ 0 and the trace of all powers of Lꢀzꢁ is zero, Lꢀzꢁ is a nilpotent operator, that is, z ꢄ ꢀψ, aꢁ is nilpotent. Thus B · I is a nil ideal. Since the Jacobson radical of an Artinian ring is the largest nilpotent ideal, we have B · JacꢀAꢁ ꢄ JacꢀAꢁ.

The last lemma, which had been proven first by Linchenko for Hopf actions and then by Nikshych for weak Hopf actions allows us to show the stability of the Jacobson radical of an algebra A which satisfies a polynomial identity and on which act some operator algebra B which is finitely generated over A. The hypotheses of the following theorem allow the reduction to finite-dimensional factors.

Theorem 2.2. Let LꢀAꢁ ⊆ B ⊆ EndkꢀAꢁ over some field k of characteristic 0 with B being finitely generated as right A -module. Suppose that for all ψ ∈ B there exist n ≥ 1 and elements

ψ11, . . . , ψm1 , ψ12, . . . , ψm2 ∈ B satisfying

  • m
  • m

  • ꢅ
  • ꢅ

  • ꢁꢂ
  • ꢃꢄ

  • L
  • ψ, a

ꢄ

ψi1 ◦ Lꢀaꢁ ◦ ψi2, ψi2 ◦ ψi1 ∈ LꢀAꢁ

Recommended publications
  • ON FULLY IDEMPOTENT RINGS 1. Introduction Throughout This Note

    ON FULLY IDEMPOTENT RINGS 1. Introduction Throughout This Note

    Bull. Korean Math. Soc. 47 (2010), No. 4, pp. 715–726 DOI 10.4134/BKMS.2010.47.4.715 ON FULLY IDEMPOTENT RINGS Young Cheol Jeon, Nam Kyun Kim, and Yang Lee Abstract. We continue the study of fully idempotent rings initiated by Courter. It is shown that a (semi)prime ring, but not fully idempotent, can be always constructed from any (semi)prime ring. It is shown that the full idempotence is both Morita invariant and a hereditary radical property, obtaining hs(Matn(R)) = Matn(hs(R)) for any ring R where hs(−) means the sum of all fully idempotent ideals. A non-semiprimitive fully idempotent ring with identity is constructed from the Smoktunow- icz’s simple nil ring. It is proved that the full idempotence is preserved by the classical quotient rings. More properties of fully idempotent rings are examined and necessary examples are found or constructed in the process. 1. Introduction Throughout this note each ring is associative with identity unless stated otherwise. Given a ring R, denote the n by n full (resp. upper triangular) matrix ring over R by Matn(R) (resp. Un(R)). Use Eij for the matrix with (i, j)-entry 1 and elsewhere 0. Z denotes the ring of integers. A ring (possibly without identity) is called reduced if it has no nonzero nilpotent elements. A ring (possibly without identity) is called semiprime if the prime radical is zero. Reduced rings are clearly semiprime and note that a commutative ring is semiprime if and only if it is reduced. The study of fully idempotent rings was initiated by Courter [2].
  • On Ring Extensions for Completely Primary Noncommutativerings

    On Ring Extensions for Completely Primary Noncommutativerings

    ON RING EXTENSIONS FOR COMPLETELY PRIMARY NONCOMMUTATIVERINGS BY E. H. FELLER AND E. W. SWOKOWSKI 0. Introduction. It is the authors' purpose in this paper to initiate the study of ring extensions for completely N primary noncommutative rings which satisfy the ascending chain condition for right ideals (A.C.C.). We begin here by showing that every completely N primary ring R with A.C.C. is properly contained in just such a ring. This is accomplished by first showing that R[x~\, x an indeterminate where ax = xa for all aeR,isN primary and then constructing the right quotient ring QCR[x]). The details of these results appear in §§1,7 and 8. The correspond- ing results for the commutative case are given by E. Snapper in [7] and [8]. If Re: A, where A is completely JVprimary with A.C.C. then,from the discussion in the preceding paragraph, it would seem natural to examine the structure of R(o) when oeA and ao = era for all aER in the cases where a is algebraic or transcendental over R. These structures are determined in §§ 6 and 8 of the present paper. The definitions and notations given in [2] will be used throughout this paper. As in [2], for a ring R, JV or N(R) denotes the union of nilpotent ideals^) of R, P or P(R) denotes the set of nilpotent elements of R and J or J(R) the Jacobson radical of R. The letter H is used for the natural homomorphism from R to R/N = R.
  • 1. Introduction. the Radical of a Ring Has Been Hitherto Defined by Using Either the Notion of a Nilpotent Or a Nil-Ideal

    1. Introduction. the Radical of a Ring Has Been Hitherto Defined by Using Either the Notion of a Nilpotent Or a Nil-Ideal

    ON THE RADICAL OF A GENERAL RING JAKOB LEVITZKI 1. Introduction. The radical of a ring has been hitherto defined by using either the notion of a nilpotent or a nil-ideal. In the following we shall ascribe the term specialized radical to the sum Na of all two- sided nilpotent ideals of a ring S, and the term generalized radical to the sum Ny of all two-sided nil-ideals of the ring. In §2 of the present note the notions of a semi-nilpotent ideal and its counterpart the semi- regular ideal are introduced, and the term radical is suggested for the sum N of all two-sided semi-nilpotent ideals of the ring. These nota­ tions may be justified by the following considerations: (a) Each nilpotent ideal is semi-nilpotent, and each semi-nilpotent ideal is a nil-ideal. (b) The radical N is a two-sided semi-nilpotent ideal which con­ tains also all one-sided semi-nilpotent ideals of the ring. (c) The radical of S/N is zero. (d) The radical N contains the specialized radical Na and is a sub­ set of the generalized radical Ny. (e) In the case of an algebra the notions: nilpotent, semi-nilpotent r r and nil-ideal are identical,andiV (r = iV = iV7;butif one turns to general rings, and replaces radical N and semi-nilpotent ideals either by specialized radical Na and nilpotent ideals or generalized radical Ny and nil-ideals, then some restriction has to be imposed on the ring 5 in order to assure the validity of (b) and (c).1 These results are applied in §3 to semi-primary rings (which will be called in short: A -rings).
  • Nil Elements and Noncommutative Rings

    Nil Elements and Noncommutative Rings

    Acta Ciencia Indica, Vol. XLIII M, No. 1 (2017) Acta Ciencia Indica, Vol. XLIII M, No. 1, 01 (2017)1 NIL ELEMENTS AND NONCOMMUTATIVE RINGS S. K. PANDEY Deptt. of Mathematics, Sardar Patel University of Police, Security and Criminal Justice, Daijar-342304, Jodhpur RECEIVED : 31 January, 2017 Nil elements (which are special type of nilpotent elements) have been introduced recently. Here we study the role of nil elements in noncommutative rings and provide some results. Among other things we prove that a noncommutative ring of order n can contain maximum n 1 nil elements and it is noted that if R be a noncommutative ring, N and N are the sets of all nil and nilpotent elements of R respectively then N is a nil ideal of R iff N is a subring of R and x ax N,x xa N, xax 0,x N,a R provided R has non-nil nilpotent elements. KEYWORDS : Even square rings, nil elements, nilpotent elements, noncommutative rings, nil ideals. AMSC2010 : 16D25, 16N40. INTRODUCTION Nil elements do not appear explicitly in the mathematical literatures [1-3] and the notion of nil elements has been introduced in [4]. It has been seen that the set of all nil elements in a commutative ring forms a nil ideal of the ring however the same is not necessarily true in the case of noncommutative rings. Recall that an element a of a ring R is called a nil element if a2 2a 0 and a ring R is called an even square ring if a2 2R,a R.
  • Lectures on Non-Commutative Rings

    Lectures on Non-Commutative Rings

    Lectures on Non-Commutative Rings by Frank W. Anderson Mathematics 681 University of Oregon Fall, 2002 This material is free. However, we retain the copyright. You may not charge to redistribute this material, in whole or part, without written permission from the author. Preface. This document is a somewhat extended record of the material covered in the Fall 2002 seminar Math 681 on non-commutative ring theory. This does not include material from the informal discussion of the representation theory of algebras that we had during the last couple of lectures. On the other hand this does include expanded versions of some items that were not covered explicitly in the lectures. The latter mostly deals with material that is prerequisite for the later topics and may very well have been covered in earlier courses. For the most part this is simply a cleaned up version of the notes that were prepared for the class during the term. In this we have attempted to correct all of the many mathematical errors, typos, and sloppy writing that we could nd or that have been pointed out to us. Experience has convinced us, though, that we have almost certainly not come close to catching all of the goofs. So we welcome any feedback from the readers on how this can be cleaned up even more. One aspect of these notes that you should understand is that a lot of the substantive material, particularly some of the technical stu, will be presented as exercises. Thus, to get the most from this you should probably read the statements of the exercises and at least think through what they are trying to address.
  • Nil and Jacobson Radicals in Semigroup Graded Rings

    Nil and Jacobson Radicals in Semigroup Graded Rings

    Faculty of Science Departement of Mathematics Nil and Jacobson radicals in semigroup graded rings Master thesis submitted in partial fulfillment of the requirements for the degree of Master in Mathematics Carmen Mazijn Promotor: Prof. Dr. E. Jespers AUGUST 2015 Acknowledgements When we started our last year of the Master in Mathematics at VUB, none of us knew how many hours we would spend on the reading, understanding and writing of our thesis. This final product as conclusion of the master was at that point only an idea. The subject was chosen, the first papers were read and the first words were written down. And more words were written, more books were consulted, more questions were asked to our promoters. Writing a Master thesis is a journey. Even though next week everyone will have handed in there thesis, we don’t yet understand clearly where this journey took us, for the future is unknown. First of all I would like to thank professor Eric Jespers, for giving me the chance to grow as mathematician in the past years. With every semester the interest in Algebra and accuracy as mathematician grew. Thank you for the guidance through all the books and papers to make this a consistent dissertation. Secondly I would like to thank all my classmates and compa˜nerosde clase. For frowned faces when we didn’t get something in class, the laughter when we realized it was a ctually quite trivial or sometimes not even at all. For the late night calls and the interesting discussions. It was a pleasure.
  • Jacobson Radical and Nilpotent Elements

    Jacobson Radical and Nilpotent Elements

    East Asian Math. J. Vol. 34 (2018), No. 1, pp. 039{046 http://dx.doi.org/10.7858/eamj.2018.005 JACOBSON RADICAL AND NILPOTENT ELEMENTS Chan Huh∗ y, Jeoung Soo Cheon, and Sun Hye Nam Abstract. In this article we consider rings whose Jacobson radical con- tains all the nilpotent elements, and call such a ring an NJ-ring. The class of NJ-rings contains NI-rings and one-sided quasi-duo rings. We also prove that the Koethe conjecture holds if and only if the polynomial ring R[x] is NJ for every NI-ring R. 1. Introduction Throughout R denotes an associative ring with identity unless otherwise stated. An element a 2 R is nilpotent if an = 0 for some integer n ≥ 1, and an (one-sided) ideal is nil if all the elements are nilpotent. R is reduced if it has no nonzero nilpotent elements. For a ring R, Nil(R), N(R), and J(R) denote the set of all the nilpotent elements, the nil radical, and the Jacobson radical of R, respectively. Note that N(R) ⊆ Nil(R) and N(R) ⊆ J(R). Due to Marks [14], R is called an NI-ring if Nil(R) ⊆ N(R) (or equilvalently N(R) = Nil(R)). Thus R is NI if and only if Nil(R) forms an ideal if and only if the factor ring R=N(R) is reduced. Hong et al [8, corollary 13] proved that R is NI if and only if every minimal strongly prime ideal of R is completely prime.
  • Γ-Semigroups with Unities and Morita Equivalence for Monoids

    Γ-Semigroups with Unities and Morita Equivalence for Monoids

    EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 6, No. 1, 2013, 1-10 ISSN 1307-5543 – www.ejpam.com Γ-semigroups with unities and Morita equivalence for monoids Sujit Kumar Sardar1,∗, Sugato Gupta1, Kar Ping Shum2 1 Department of Mathematics, Jadavpur University,India 2 Institute of Mathematics, Yunnan University, China Abstract. In this paper we show that the left operator and right operator semigroups of a Γ-semigroup with unities are Morita equivalent monoids. It is also deduced that if L and R are two Morita equivalent monoids then a Γ-semigroup with unities can be constructed such that its left and right operator monoids are isomorphic to L and R respectively. This result is used to obtain some properties of monoids which remain invariant under Morita equivalence. 2010 Mathematics Subject Classifications: 20M11, 20M12, 20M30, 20M50. Key Words and Phrases: Γ-semigroup, Morita equivalence for monoids, Morita invariant. 1. Introduction If someone asks "What is the most natural example of ring?", the ring of integers is one possible answer to this question. But a more interesting answer will be the endomorphism ring of an abelian group, i.e., End M or Hom(M, M) where M is an abelian group. Now if two abelian groups, say A and B instead of one are taken, then Hom(A, B) is no longer a ring in the way as End M becomes a ring because the composition is no longer defined. However, if one takes an element of Hom(B, A) and put it in between two elements of Hom(A, B) then the composition can be defined.
  • MATH 210C: HOMEWORK 5 Problem 40. Recall That the Nilradical of A

    MATH 210C: HOMEWORK 5 Problem 40. Recall That the Nilradical of A

    MATH 210C: HOMEWORK 5 Problem 40. Recall that the nilradical of a commutative ring A is the set of all nilpotent elements, that is, all x 2 A such that xn = 0 for some n 2 N. (a) Prove that the nilradical is the intersection of all prime ideals, and hence itself an ideal. Prove that the nilradical is always contained in the Jacobson radical. (b) Let A = k[X1;:::;Xn]=I be a finitely generated k-algebra for k a field. Prove that its nilradical and its Jacobson radical coincide. (c) Let A = R[[X]], the formal power series ring of a Noetherian commutative ring R. Compute the nilradical and the Jacobson radical of A, and conclude that these ideals are not equal in general. Problem 41. If A is a noncommutative ring, we can still consider the subset of nilpotent elements of A. Prove that this subset is generally neither closed under addition nor under left or right multiplication by elements of A. Further, give an example of two nilpotent elements whose product is no longer nilpotent. Problem 42. Let A be a ring with nilradical N. Suppose that every ideal I not contained in N contains a nonzero idempotent. Prove that Jac(A) = N. Problem 43. Compute the Jacobson radical of Z. Problem 44. Compute the Jacobson radical of Z=24Z. Generally, what is the Jacobson radical of Z=nZ? Problem 45. Let e 2 A be a central idempotent, i.e. it belongs to the center of A and satisfies e2 = e. Let M be a left A-module.
  • Some Characterizations of Finite Commutative Nil Rings Michael Holloway

    Some Characterizations of Finite Commutative Nil Rings Michael Holloway

    Palestine Journal of Mathematics Vol. 2(1) (2013) , 6–8 © Palestine Polytechnic University-PPU 2013 Some Characterizations of Finite Commutative Nil Rings Michael Holloway Communicated by Salah-Eddine Kabbaj MSC 2010 Classifications: Primary: 16N40; Secondary: 13A15. Keywords and phrases: nil ring, nil ideal, irreducible ideal, zero-divisor. This article is based on the author’s master’s thesis at Tennessee Technological University. Thanks to Dr. Andrew J. Hetzel, the author’s major advisor, for his help and guidance. The author wishes to thank the referee for his revisions of this article, especially in regards to the creation of a much shorter proof of Theorem 1. Abstract. In this article, some ideal-theoretic characterizations are provided concerning when a commutative ring is nil under certain finiteness conditions of the ring. It is shown that if R is a finitely generated ring without unity, then R is nil if and only if every proper ideal of R is nil. It is also shown that if R is a finite ring with every proper ideal irreducible, then R is nil if and only if every element of R is a zero-divisor. 1 Introduction Throughout this article, all rings are commutative and do not necessarily have unity. A ring is called nil if every element of the ring is nilpotent; that is, the ring R is nil if for every element x of R, there is a positive integer n for which xn = 0. More strongly, the ring R is called nilpotent if there is a positive integer m such that Rm = f0g.
  • Non-Nil Ideal Radical and Non-Nil Noetherian R[X]

    Non-Nil Ideal Radical and Non-Nil Noetherian R[X]

    Middle-East Journal of Scientific Research 15 (12): 1663-1665, 2013 ISSN 1990-9233 © IDOSI Publications, 2013 DOI: 10.5829/idosi.mejsr.2013.15.12.431 Non-nil Ideal Radical and Non-nil Noetherian R[x] M. R. Alinaghizadeh and R. Modabbernia Department of Mathematics, Shoushtar Branch, Islamic Azad University, Shoushtar, Iran Abstract: In the part two of this paper we investigate Non-Nil ideal radical. In the part three we show that if R non-nil noetherian ring, Nil(R) divided prime ideal and nil(R[x])Í(f)"fÎR[x]\nil(R[x]). Then R[x] Non-nil noetherian Ring. Key words : Non-nil radical ideal · non-nil noetherian · divided prime ideal · minimal prime ideal INTRODUCTION (3)?(1) Suppose I be a non-nil ideal of R and F = {(a1, a2,… am)| mÎN, aiÎI, (a1,… am) non-nil ideal}. We suppose that all of the rings in this article are Since I is non-nil thus $aÎl such that aÏnil(R), commutative with 1¹0. (a)ÎF, F¹Æ. So according to (3) F has maximal Let R be a ring. Then nil(R) denotes the set of element (a1,..,am) we will show that I = (a1,..,am). nilpotent elements of R.I is called non-nil ideal of R if If I¹(a1,..,am) then $aÎI\(a1,..,am) thus I and {rÎR|$nÎN, rnÎI}. J is a non-nil (a1,..,am) ,…..,am,a) and (a1,..,am,a)ÎF we get to ideal radical of R if $ non-nil ideal I J P is divided a contradiction. prime ideal if pÍ(x)"xÎR\P.
  • On Nonnil-S-Noetherian Rings

    On Nonnil-S-Noetherian Rings

    mathematics Article On Nonnil-S-Noetherian Rings Min Jae Kwon and Jung Wook Lim* Department of Mathematics, College of Natural Sciences, Kyungpook National University, Daegu 41566, Korea; [email protected] * Correspondence: [email protected] Received: 27 July 2020; Accepted: 24 August 2020; Published: 26 August 2020 Abstract: Let R be a commutative ring with identity, and let S be a (not necessarily saturated) multiplicative subset of R. We define R to be a nonnil-S-Noetherian ring if each nonnil ideal of R is S-finite. In this paper, we study some properties of nonnil-S-Noetherian rings. More precisely, we investigate nonnil-S-Noetherian rings via the Cohen-type theorem, the flat extension, the faithfully flat extension, the polynomial ring extension, and the power series ring extension. Keywords: nonnil-S-Noetherian ring; S-Noetherian ring; S-finite ideal; SFT ring MSC: 13A15; 13B25; 13E99; 13F25 1. Introduction The concept of Noetherian rings is one of the most important topics that is widely used in many areas including commutative algebra and algebraic geometry. The Noetherian property was originally due to the mathematician Noether who first considered a relation between the ascending chain condition on ideals and the finitely generatedness of ideals. More precisely, she showed that if R is a commutative ring with identity, then the ascending chain condition on ideals of R holds if and only if every ideal of R is finitely generated. This equivalence plays a significant role in simplifying the ideal structure of a ring. Due to the importance of Noetherian rings, many mathematicians have tried to use Noetherian properties in several classes of rings and attempted to generalize the notion of Noetherian rings.