Faithful Module Research Articles

A Short Note on the Schur-Jacobson Theorem

An upper bound for the length of a commutative Artinian ring in terms of the length of a faithful module is given. This generalizes a theorem of Schur and Jacobson.

A short note on the Schur-Jacobson theorem

An upper bound for the length of a commutative Artinian ring in terms of the length of a faithful module is given. This generalizes a theorem of Schur and Jacobson.

Defeat of the 𝐹𝑃²𝐹 conjecture: Huckaba’s example

A commutative ring R R is F P 2 F F{P^2}F (resp. FPF) provided that all finitely presented (resp. finitely generated) faithful modules generate the category mod − R \bmod - R of all R R -modules. A conjecture of the author dating to the middle 1970s states that any F P 2 F F{P^2}F ring R R has FP-injective classical quotient ring Q = Q c l ( R ) Q = {Q_{cl}}(R) . It was shown by the author (Injective quotient rings. II, Lecture Notes in Pure and Appl. Math., vol. 72, Dekker, New York, 1982, pp. 71-105) that FPF rings R R have injective Q Q and by the author and P. Pillay (Classification of commutative FPF rings, Notas Math., vol. 4, Univ. de Murcia, Murcia, Spain, 1990) that C F P 2 F CF{P^2}F local rings (defined below) have FP-injective Q Q . The counterexample is a difficult example due to Huckaba of a strongly Prüfer ring without "Property A." (A ring with Property A was labelled a McCoy ring by the author.) This counterexample is C F P 2 F CF{P^2}F in the sense that every factor ring of R R is F P 2 F F{P^2}F .

QF-1 algebras with faithful direct sums of uniserial modules

Let M be a left unital module over a ring R. The module M is called balanced if the canonical ring homomorphism from R into the double centralizer of M is surjective. Following Thrall [lOI R is said to be QF-1 if every finitely generated faithful R-module is balanced. In [6] Ringel proved that if an algebra with square zero radical is QF-1, then it is a Tachikawa algebra (= an algebra of local-colocal type), i.e., an algebra each of whose indecomposable modules has either the simple top or the simple socle. Also, Tachikawa [9] proved that if an algebra whose left regular module is serial (= a left serial algebra) is QF-1, then it is a Tachikawa algebra (particularly, an algebra of left colocal type). Here, a serial module is a direct sum of uniserial modules (a module is uniserial in case it has the unique composition series). It should be noted that an algebra with square zero radical, as well as a left serial algebra, has a faithful serial module. The purpose of this paper is to give a generalization of both of the above results by Ringel and by Tachikawa in the case where the ground field of an algebra is algebraically closed. Indeed we shall prove

On Irreducible Faithful Modules and their Cohomology

Bulletin of the London Mathematical SocietyVolume 23, Issue 1 p. 75-77 Notes and papers On Irreducible Faithful Modules and their Cohomology W. Willems, W. Willems Institute of Experimental Mathematics, University of Essen, Ellernstrasse 29, 4300 Essen 12, West GermanySearch for more papers by this author W. Willems, W. Willems Institute of Experimental Mathematics, University of Essen, Ellernstrasse 29, 4300 Essen 12, West GermanySearch for more papers by this author First published: January 1991 https://doi.org/10.1112/blms/23.1.75AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat No abstract is available for this article. Volume23, Issue1January 1991Pages 75-77 RelatedInformation

Lie algebra representations of dimension <𝑝²

Various methods of representation theory of modular Lie algebras are improved. As an application the structure of the Lie algebras having a faithful irreducible module of dimension > p 2 > {p^2} is determined. Applications to the classification theory of modular simple Lie algebras are given.

A Decomposition of Rings Generated by Faithful Cyclic Modules

AbstractA ring R is said to be generated by faithful right cyclics (right finitely pseudo-Frobenius), denoted by GFC (FPF), if every faithful cyclic (finitely generated) right R-module generates the category of right R-modules. The class of right GFC rings includes right FPF rings, commutative rings (thus every ring has a GFC subring - its center), strongly regular rings, and continuous regular rings of bounded index. Our main results are: (1) a decomposition of a semi-prime quasi-Baer right GFC ring (e.g., a semiprime right FPF ring) is achieved by considering the set of nilpotent elements and the centrality of idempotnents; (2) a generalization of S. Page's decomposition theorem for a right FPF ring.

On rings whose finitely generated faithful modules are generators

A ring R over which every finitely generated faithful right R-module is a generator of the category mod-R of all right R-modules is called right FPF. FPF ring theory was initiated in this general setting by Faith in order to study from a unified point of view those rings that appear in Morita duality, commutative Prtifer rings and bounded Dedekind prime rings amongst others. The reader can consult Faith and Page’s book [ 141, where most of the basic results on FPF rings are contained. In Section 1 of this paper we construct semiprime FPF rings that are not semihereditary, thus answering [ 14, Question 111. It was an open question [l, p, 173171 whether or not the Pierce stalks of a semiprime FPF ring are FPF. We will answer this in the negative. We close Section 1 by proving that the maximal ring of quotients of a semiprime right FPF ring R satisfying a polynomial identity is the localization of R at the set of all nonzero divisors of the centre Z of R. In the case where R is module finite over Z this was obtained in [l, Proposition 1.91. Section 2 considers centres, Galois subrings, and group rings over FPF rings. By using the methods of Bergman and Cohn, cf. [4, Section 6.21, we show that every integrally closed commutative domain can be realized as the centre of a Bezout FPF domain, and conversely. This answers [14, Question 33 in the negative. We remark that a negative answer for nonsemiprime FPF rings is implicit in [24], where the authors construct a quasi-Frobenius ring whose centre is not quasi-Frobenius. We also provide additional examples showing that Galois subrings of semiprime FPF rings need not be FPF, cf. [14, Question 141. A positive result is that if C is a finite group of automorphisms of a reduced commutative FPF ring, then the fixed ring RG is FPF; this is fairly easy but our examples illustrate and limit the scope for possible generalizations. 425

Tame Objects for Finite Commutative Hopf Algebras

Let $S$ be an $A$-module algebra for a commutative Hopf algebra $A$, both projective of the same rank over a commutative ring. Let ${\mathbf {I}}$ be the space of integrals in $A$. Then $S$ is an invertible $A$-module iff it is a faithful module which satisfies the "trace surjectivity" condition that 1 is in ${\mathbf {I}}S$.

Rings with projective socle

The class of rings with projective left socle is shown to be closed under the formation of polynomial and power series extensions, direct products, and matrix rings. It is proved that a ring R R has a projective left socle if and only if the right annihilator of every maximal left ideal is of the form f R fR , where f f is an idempotent in R R . This result is used to establish the closure properties above except for matrix rings. To prove this we characterise the rings of the title by the property of having a faithful module with projective socle, and show that if R R has such a module, then so does M n ( R ) {M_n}\left ( R \right ) . In fact we obtain more than Morita invariance. Also an example is given to show that e R e eRe , for an idempotent e e in a ring R R with projective socle, need not have projective socle. The same example shows that the notion is not left-right symmetric.

Tame objects for finite commutative Hopf algebras

Let S S be an A A -module algebra for a commutative Hopf algebra A A , both projective of the same rank over a commutative ring. Let I {\mathbf {I}} be the space of integrals in A A . Then S S is an invertible A A -module iff it is a faithful module which satisfies the "trace surjectivity" condition that 1 is in I S {\mathbf {I}}S .

Semi-perfect FPF rings and applications

A ring R is right FPF if each faithful finitely generated right R module is a generator of MOD- R. The purpose of this paper is to answer several questions posed by C. Faith and S. Page in their monograph (“FPF Ring Theory: Faithful modules and generators of MOD- R,” London Math. Soc. Lecture Note Series, Cambridge Univ. Press, 1984), and then to apply these results to semi-perfect left or right Noetherian right FPF rings. Thus, a semi-perfect ring R is right FPF iff (i) R possesses a semi-perfect, right self-injective classical (left and right) ring of quotients, (ii) faithful finitely generated right ideals of R are generators of MOD- R, and (iii) the basic ring of R is right strongly bounded. Furthermore, the set of left regular elements of R is the set of right regular elements of R is the set of elements regular modulo Z r ( R). The semi-perfect ring R is an injective cogenerator of MOD- R iff R is right FPF and right Kasch. Chain conditions on R are also considered.

Quotient Rings of Rings Generated by Faithful Cyclic Modules

A ring $R$ is said to be generated by faithful cyclics (right finitely pseudo-Frobenius), denoted by right GFC (FPF), if every faithful cyclic (finitely generated) right $R$-module generates the category of right $R$-modules. A fundamental result in FPF ring theory, due to S. Page, is that if $R$ is a right nonsingular right FPF ring, then ${Q_r}(R)$ is FPF. In this paper we generalize this result by providing a necessary and sufficient condition for a right nonsingular right GFC ring to have an FPF maximal right quotient ring.

On modules with DICC

Let R be an arbitrary commutative ring with unit. R-modules with DICC, i.e., modules which have no doubly infinite chain of submodules, were first introduced in [Cl] and extensively investigated in [C2]. Let us recall from those papers some results which will be referred to later on, but first let us fix some notation. If R is a commutative ring with unit, n will denote its nilradical. Also, a min/max ideal of R is a prime ideal which is both minimal and maximal. If M is an R-module, then N(M) = N = the sum of all submodules of M of finite length. In [Cl] we proved that a non-reduced ring S with no min/max ideals has DICC if and only if S reduced is Noetherian, and n has DCC, is nilpotent, and is almost divisible (i.e., Vyg Sn, n/yn has finite length). In [C2] we proved a similar result for modules--that is, an R-module M has DICC if and only if N has DCC; M/N has ACC; and, for all x E M N, N/Rx n N has finite length. Moreover, a DICC not DCC R-module M can be generated by finitely many elements outside the submodule N. In this paper we make a further investigation of DICC modules. We call weakly DZCC (abbreviated to WDICC a ring whose nilradical is nilpotent and DCC, and such that the reduced ring is Noetherian; we call standardly DZCC (abbreviated to SDICC) a DICC module (resp. ring) which is not ACC, not DCC (resp. not ACC and, hence, also not DCC). We pay particular attention to the question of how the existence of a DICC R-module affects the ring itself. It is known that if a ring admits a faithful ACC module, then the ring is Noetherian. Recently, W. Heinzer and D. Lantz proved that if a complete quasi-local ring admits a faithful DCC module, then it is Noetherian. For SDICC modules there seems to be no obvious way of proving a similar result, as the Example 1 given after Theorem 1 shows. In this subtle example we exhibit a quasi-local WDICC ring R such that Rred is complete local, R has a faithful SDICC module, but R is not DICC. 75 0021-8693/87 $3.00

Quotient rings of rings generated by faithful cyclic modules

A ring R R is said to be generated by faithful cyclics (right finitely pseudo-Frobenius), denoted by right GFC (FPF), if every faithful cyclic (finitely generated) right R R -module generates the category of right R R -modules. A fundamental result in FPF ring theory, due to S. Page, is that if R R is a right nonsingular right FPF ring, then Q r ( R ) {Q_r}(R) is FPF. In this paper we generalize this result by providing a necessary and sufficient condition for a right nonsingular right GFC ring to have an FPF maximal right quotient ring.

Representation of rings with faithful polyform modules

This article studies subrings which satisfy a density-type criterion called m-density. It is first observed that if V is a faithful quasiinjective R-module then R is an m-dense subring of BiendRV. This is used to obtain the main result which states that R is an m-dense subring of a direct product of rings of linear transformations if and only if R has a faithful locally finite dimensional module whose essential sub-modules are rational.

QF-1Algebras which have infinitely many minimal faithful modules

QF-1Algebras which have infinitely many minimal faithful modules

ℵ-QF 3 rings with zero singular ideal

In 1948 R. M. Thrall [ 143 introduced several generalizations of the concept of a Quasi-Frobenius algebra over a field. In particular, he called an algebra R (right) QF 3 if there is a minimal faithful right R-module U, in the sense that if a module M, is faithful, then M= U@ N for an appropriate module N,. Since then, many authors studied rings satisfying that condition, calling them right QF 3 rings. However, it should be noted that Thrall’s original definition of a right QF 3 ring, applied to an arbitrary ring, was considered for the first time in 1969 by Colby and Rutter [3]. Indeed, in all previous papers, the authors were concerned with right QF 3 rings satisfying some additional conditions, such as being (right and left) artinian or, at least, semiprimary. One of the basic results obtained by Colby and Rutter is the following (see [3, Theorem 11): a ring R (associative with unit) is a right QF 3 ring if and only if there is a finite family (en)A,n of pairwise orthogonal idempotents of R, with e,R & e, R if II # ,u, such that each eA R is the injective envelope of a minimal right ideal and W= xi e,R is a faithful right ideal; if it is the case, then W, is a minimal faithful module, as explained above. The natural extension of the concept of a right QF 3 ring, which consists in casting out the finiteness condition on the set A, was considered by Y. Kawada in [lo]: he calls R a right N-QF 3 ring if there is a family (eAsn of idempotents of R satisfying the above conditions. Here H stands for the cardinality of A, so that a right QF 3 ring is nothing else than a right N-QF 3 ring with K a finite cardinal. Right N-QF 3 rings share with right QF 3 rings several known properties and Kawada gave in his paper a way to build examples of such rings (see [lo, Theorem 5.73). The purpose of the present paper is the study of right N-QF 3 rings with zero singular right ideal. The results that we shall obtain generalize some

The arens closure of a nonassociative algebra

Given a nonassociative algebra A and an Arens pair A1, A2, for A, we identify a subalgcbra [Abar] of A2 with i (A) ⊂ A ⊂ A2 and show that [Abar] better reflects the algebraic structure ot A, in parti-cular. any multilinear identity satisfied by [Abar] is also satisfied by [Abar] Hence, [Abar] is commutative or Lie when A is and Jordan when A is a Jordan algebra of characteristic not 2 or 3. Also, we list examples (1) where [Abar] = EndD(V) for A a primitive, associative algebra with commuting ring D and irreducible faithful module V,(2) where [Abar] is the norm closure of A in the arens algebra of all bounded functionals of the bounded functionals for a normed algebra A and (3) where [Abar] is the Arens algebra of all bounded functionals of the bounded functionals with A again normed. Note that dif-ferent Arens closures can arise form the same choice of A, A1, , A2 since [Abar] is determined by A, A1, A2 and subspaces A3 ⊂ A2 *, A4,⊂A3 *.

Representations under ring extensions: Latimer-MacDuffee and Taussky correspondences

C. G. Latimer and C. C. MacDuffee defined in [8] a bijective correspondence between similarity classes of nonsingular rational integral n x n matrices having a common separable characteristic polynomialf(x) and the classes of faithful ideals in the ring Z[x]/f(x) Z[x], Z the ring of rational integers. Several authors have used this simple duality with the ideas indigenous to one theory to obtain valuable consequences in the other (see [14] and [15] f or a survey), thereby motivating further analysis of the correspondence (as in [ 1, 2, 4, 17 and 181). Nevertheless, the theory has generally been catagorized as a subset of module theory of orders in semisimple algebras following Zassenhaus’s proof of Jordan’s theorem on the finiteness of such similarity classes (see [ 7, 14 and 21 I). The purpose of this note is to replace the semisimple assumption present in these previous studies with the weaker condition that the modules considered become trivial under a suitable extension of the base ring. The ease in which matrix and ideal theory are transferable under the Latimer-MacDuffee duality prompted our adherence to their approach, when possible, in this development over arbitrary commutative rings. We review prior matrix/ideal correspondences in Section 2 preliminary to the general framework in Section 3. There an n x II matrix class is replaced by the isomorphism class of a rank n projective R module M which is also a faithful module over a rank n projective R algebra A. The ideal theory is obtained by extending the base ring R of scalars to a commutative ring S for which M becomes trivial (S @ll MN S OR A as S OR A modules), and M is then identified with a fractional A ideal in S OR A. In the classical case, S is the field of quotients of a domain R. Theorem 1 reduces to the Latimer-MacDuffee duality in the latter case. Theorem 2 shows that the theory is equally adaptable to any commutative rank IZ projective R algebra A which is isomorphic to its dual over a suitable extension of R. As an application, we show that a nonderogatory matrix is similar to the companion matrix of its