Modules whose endomorphism rings have isomorphic maximal left and right quotient rings

This paper investigates two related problems. The first is to describe the double centralizer of an arbitrary projective right iϋ-modiile. This proves to be the ring of left quotients of R with respect to a certain canonical hereditary torsion class of left ϋί-modules determined by the projective module. The second is to determine the relationship between rings of left quotients of R and S, where S is the endomorphism ring of a finitely generated projective right ϋί-module PR. It is shown that there exists an inclusion-preserving, one-to-one correspondence between hereditary torsion classes (or localizing subcategories) of left ^-modules and hereditary torsion classes of left E-modules which contain the canonical torsion class determined by PR. If QR and Qs are rings of left quotients with respect to corresponding classes, then P($$RQR is a finitely generated projective right Q^-module with Qs as its Q#-endomorphism ring. Necessary and sufficient conditions are obtained for the maximal rings of left quotients to be related in this manner. In particular, this occurs when PR is a faithful .R-module and R is either a semi-prime ring or a ring with zero left singular ideal. The situation considered includes the case where S is an arbitrary ring, SP is a left ^-generator, and R is the Sendomorphism ring of SP When SP is a projective left Sgenerator, the maximal rings of left quotients of R and S are related in the manner considered above.

Let a be an idempotent kernel functor defining the ring of left quotients Q"(R).We introduce a notion of cr-divisibility, and show that a cr-torsionfree /{-module M is a module over Qa(R) if and only if M is cr-divisible.In his paper Rings and modules of quotients, Goldman [2] investigated the construction of rings of quotients of a ring R with respect to idempotent kernel functors of the category RM of left -modules.If a is such an idempotent kernel functor, then it was shown that every cr-torsionfree and o--injective left -module is a left module over the ring of quotients Qa(R).We have called a module RM o--divisible if it is generated in the categorical sense by the A-module Qa(R).In Theorem 4 below we show that a atorsionfree module RM is a left ff(F)-module (extending the action of R) if and only if M is cr-divisible.In Theorem 6 we show that the quotient functor Q" is naturally isomorphic to the functor Qa(R)_if and only if every ff-torsionfree and cr-divisible left ^-module is cr-injective.All rings considered in this note are associative rings with identity, and all modules are assumed to be unital.For the convenience of the reader we repeat the relevant definitions of [2] and [3].A functor o from RM into itself is called a kernel functor if it has the following properties: (i) a(M) is a submodule of M for every M e RM. (ii) Iff: M->-N is a homomorphism, then f(o(M))^ o(N), and o(f) is the restriction off to o(M).(iii) If M' is a submodule of M-, then a(M') = M' t~\a(M).If, in addition, o(Mo(M))=0for all M e RM, then a is called an idempotent kernel functor.For the remainder of the note we let a be a fixed idempotent kernel functor of RM.A module RM is called a-torsion if o(M) = M; it is called cr-torsionfree if o(M)=0.A submodule M' of M is said to be cr-dense in M if M\M' is cr-torsion ; it is said to be tr-closed in M if M\M' is cr-torsionfree.The intersection of all cr-closed submodules of M which contain M' is called the o--closure of M' in M.

In a recent paper2 we have called a ring 2 primitive if 2 contains a maximal right ideal a such that the quotient (a: W) = (0). The quotient (: 1) is defined to be the largest two-sided ideal of 21 that satisfies the condition 21(a: 21) C a. Thus if 21lias an identity, (a: 21) is the largest two-sided ideal of 21 contained in a. Primitive rings appear to play a role in the general structure theory of rings analogous to that played by simple rings in the Wedderburn-Artin theory. Primitive rings are also fundamental in general representation theory since it is known that a necessary and sufficient condition that a ring 21 be primitive is that 21 be isomorphic to an irreducible ring W of endomorphisms in a commutative group T. If 2 is irreducible in T and Zi is the division ring of endomorphisms commutative with the elements of A, then R is a dense ring of linear transformations in T regarded as a vector space over Z.3 We shall call 2 a left primitive ring if 2 contains a maximal left ideal a' such that the quotient (}': 21)l = (0). Here (s': 2) 1 is the largest two-sided ideal in 2 having the property (a': 21)1 21 C a'. Evidently, a ring is left primitive if and only if it is anti-isomorphic to an irreducible ring of endomorphisms. It is an open question whether or not a primitive ring is necessarily left primitive. This is indeed the case if 2 contains minimal ideals. In many other respects the theory of primitive rings that contain minimal ideals constitutes the most satisfactory part of the theory of primitive rings. An appropriate tool for studying rings of this type is a generalization of the duality theory previously used by Dieudonn6 in studying simple rings that possess minimal ideals.4 In this note we develop this generalization of Dieudonn6's results. We also investigate certain natural topologies that can be defined in any primitive ring.

In this paper, we show how to reduce the study of nondegenerate local Goldie (-1, 1) rings to the strongly prime case, via the notions of uniform ideals and essential subdirect product. Also, we construct the maximal left quotient ring of (-1, 1) ring that is a left quotient ring of itself. We follow Utumi where a maximal left quotient ring is constructed as a direct limit of a partially defined homomorphism from the left ideal of R to R.

A left module M over a ring R is cofaithful in case there is an embedding of R into a finite product of copies of M. Our main result states that a semiprime ring R is left Goldie, that is, has a semisimple Artinian left quotient ring, if and only if R satisfies (i) every faithful left ideal is cofaithful and (ii) every nonzero left ideal contains a nonzero uniform left ideal. The proof is elementary and does not make use of the Goldie and LesieurCroisot theorems. We show that (i) and (ii) are Morita invariant. Moreover, (ii) is invariant under polynomial extensions, and so is (i) for commutative rings. Absolutely torsionfree rings are studied. The ring Q is a left classical quotient ring for the ring R C Q if every regular element (nondivisor of zero) of R is invertible in Q and if every element of Q is of the form b ~ιa where a,b E R and b is regular; in this case we also say that R is a left order in Q. A ring is said to be left Goldie if it has the ascending chain condition on left annihilators and has finite uniform dimension. (A left R -module has finite uniform dimension if it has no infinite direct sum of nonzero submodules, and it is said to be uniform if it is nonzero and any two nonzero submodules have a nontrivial intersection.) A theorem of Goldie [8,9] and Lesieur and Croisot [12] states that a ring is a left order in a semisimple Artinian ring if and only if it is semiprime and left Goldie. It is known that the ascending chain condition on left annihilators is not preserved under an equivalence of categories (Morita invariant); in fact, it does not go up to matrix rings. It is unknown whether being left Goldie is Morita invariant. In section two we give a proof of the theorem stated in the abstract, and in the prime case we give a proof which shows directly that such a ring is an order in a full matrix ring over a division ring. We also weaken the hypothesis of an important theorem on semiprime PI rings. In the third section we use these techniques to study absolutely torsion-free rings. In particular, we show that an absolutely torsionfree ring is Goldie if and only if it has a uniform left ideal, and that the endomorphism ring of a finitely generated projective module over an absolutely torsion-free ring is absolutely torsion-free.

This series aims to report new developments in mathematical research and teaching -quickly, informally and at a high level.The type of material considered for publication includes: 1.Preliminary drafts of original papers and monographs 2. Lectures on a new field, or presenting a new angle on a classical field 3. Seminar work-outs 4. Reports of meetings, provided they are a) of exceptional interest or b) devoted to a single topic.

For $R$ a subring of an $n \times n$ matrix ring ${M_n}(\Delta )$ over a division ring $\Delta$, we examine an invariant called the row rank of $R$. Roughly speaking, the row rank of $R$ is the largest integer $k$ such that $R$ contains all $k$-rowed matrices over a left order in $\Delta$. The row rank of $R$ is then an integer between 0 and $n$; and we will see that row rank $R \geq 1$ means that ${M_n}(\Delta )$ is the maximal left quotient ring of $R$, while row rank $R = n$ signifies that ${M_n}(\Delta )$ is the classical left quotient ring of $R$. Thus row rank provides a link between maximal and classical quotient rings for rings of this type. A description of the subrings $R$ with row rank $R \geq k$ is obtained which subsumes and generalizes earlier theorems of Faith-Utumi and Zelmanowitz, respectively, for the cases row rank $R = n$ and row rank $R \geq 1$.

Centers and prime ideals in group algebras of polycyclic-by-finite groups

This paper consists of three parts. The first is devoted to investigating the equivalence and left-right symmetry of several conditions known to characterize finite dimensional algebras which have a unique minimal faithful representation— QF- 3 3 algebras—in the class of left perfect rings. It is shown that the following conditions are equivalent and imply their right-hand analog: R R contains a faithful Σ \Sigma -injective left ideal, R R contains a faithful I I II -projective injective left ideal; the injective hulls of projective left R R -modules are projective, and the projective covers of injective left R R -modules are injective. Moreover, these rings are shown to be semi-primary and to include all left perfect rings with faithful injective left and right ideals. The second section is concerned with the endomorphism ring of a projective module over a hereditary or semihereditary ring. More specifically we consider the question of when such an endomorphism ring is hereditary or semihereditary. In the third section we establish the equivalence of a number of conditions similar to those considered in the first section for the class of hereditary rings and obtain a structure theorem for this class of hereditary rings. The rings considered are shown to be isomorphic to finite direct sums of complete blocked triangular matrix rings each over a division ring.

While the classical right ring of quotients Q cl r (R) exists (if and) only if R is a right Ore ring, a maximal right ring of quotients Qmaxr(R) exists for any ring R. To define Q max r (R), one must first understand the structure of the endomorphism ring of a QI (quasi-injective) module.

An associative ring R with identity is said to be a left FTF ring when the class of the submodules of flat left R-modules is closed under injective hulls and direct products .We prove (Theorem 3.5) that a strongly graded ring R by a locally finite group G is left FTF if and only if Re is left FTF, where e is the neutral element of G.This provides new examples of left FTF rings.Some consequences of this Theorem are given. . IntroductionIn the papers [5], [6] we started the study of the rings R with the property that the class of the submodules of flat left modules, .PO, is closed under direct products and injective hulls.For these rings, .Po is the torsionfree class for some hereditary torsion theory on R-Mod.Thus we can use techniques of localization relative to this torsion theory to investigate these rings.This class of rings is large and we call them left FTF rings.Clearly, QF and regular rings are left and right FTF rings .In the commutative case, Enochs showed [3] that a noetherian commutative ring R is FTF if and only if Rp is Gorestein for every minimal prime ideal P of R. Noncommutative examples of left FTF rings can be found in [5], [6] .The aim of this note is to construct new examples of left FTF rings .The tool will be the strongly graded ring theory.The paper is organized as follows .In Section 1 we record the fundamental results on left FTF rings showed in [5] and [6] which we will use in the rest of the paper.We recall also the basic machinery from torsion theories and graded rings that we will use .The main result, Theorem 2.5, appear in Section 2 .Finally, we give some applications of this result in Section 3.

Let B be the ring of integral valued polynomials over a noetherian domain A. We study in which case finitely generated ideals of B are uniquely determined by their ideals of values at each element of A. We give necessary and sufficient conditions which are verified for example when A is any ring of integers of an algebraic number field, such that each quotient ring Am with respect to a maximal ideal m is analytically irreducible.

In this article, the index of imprimitivity of an irreducible nonnegative matrix in the famous Perron-Frobenius theorem is studied within a more general framework, both in a more general tensor setting and in a more natural spectral symmetry perspective. A $k$-th order tensor has symmetric spectrum if the set of eigenvalues is symmetric under a group action with the group being a subgroup of the multiplicative group of $k$-th roots of unity.A sufficient condition, in terms of linear equations over the quotient ring, for a tensor possessing symmetric spectrum is given, which becomes also necessary when the tensor is nonnegative, symmetric and weakly irreducible, or an irreducible nonnegative matrix. Moreover, it is shown that for a weakly irreducible nonnegative tensor, the spectral symmetries are the same when either counting or ignoring multiplicities of the eigenvalues.In particular, the spectral symmetry (index of imprimitivity) of an irreducible nonnegative Sylvester matrix is completely resolved via characterizations with the indices of its positive entries. It is shown that the spectrum of an irreducible nonnegative Sylvester matrix can only be $1$-symmetric or $2$-symmetric, and the exact situations are fully described. With this at hand, the spectral symmetry of a nonnegative two-dimensional symmetric tensor with arbitrary order is also completely characterized.

In her paper [3], Osofsky exhibited an example of a ring R containing 16 elements which (i) is equal to its left complete ring of quotients, (ii) is not self-injective and (iii) whose injective hull HR = H(RR) allows a ring structure extending the R-module structure of HR. In the present note, we offer a general method of constructing such rings; in particular, given a non-trivial split Frobenius algebra A and a natural n ≧ 2, a certain ring of n x n matrices over A provides such an example. Here, taking for A the semi-direct extension of Z/2Z by itself and n = 2, one gets the example of Osofsky. Thus, our approach answers her question on finding a non-computational method for proving the existence of such rings.

A left artinian ring Λ is projectively stable if no non-zero morphism f: M → N of finitely generated left Λ-modules M,N having no non-zero projective direct summands factors through a projective module. Such rings have many good properties and, although the definition is given in terms of the category of modules, the structure of projectively stable left artinian rings allows a satisfying description: they are “built” from well-known left artinian rings, left hereditary and serial, by a pull back of rings construction. We show that representations of projectively stable artin algebras are also “built” from representations of well-studied artin algebras, hereditary and serial. Namely, we give a simple geometric construction that produces the Auslander-Reiten quiver or the ordinary quiver of a projectively stable algebra from those of an hereditary algebra and a serial algebra.