Finitely Presented Modules Over Semihereditary Rings

This paper investigates the structure of cyclically pure (or C-pure) projective modules. In particular, it is shown that a ring R is left Noetherian if and only if every C-pure projective left R-module is pure projective. Also, over a left hereditary Noetherian ring R, a left R-module M is C-pure projective if and only if M = N ⊕ P, where N is a direct sum of cyclic modules and P is a projective left R-module. The relationship C-purity with purity and RD-purity are also studied. It is shown that if R is a local duo-ring, then the C-pure projective left R-modules and the pure projective left R-modules coincide if and only if R is a principal ideal ring. If R is a left perfect duo-ring, then the C-pure projective left R-modules and the pure projective left R-modules coincide if and only if R is left Köthe (i.e., every left R-module is a direct sum of cyclic modules). Also, it is shown that for a ring R, if every C-pure projective left R-module is RD-projective, then R is left Noetherian, every p-injective left R-module is injective and every p-flat right R-module is flat. Finally, it is shown that if R is a left p.p-ring and every C-pure projective left R-module is RD-projective, then R is left Noetherian hereditary. The converse is also true when R is commutative, but it does not hold in the noncommutative case.

Commutative Noetherian local rings whose ideals are direct sums of cyclic modules

In this paper, we introduce and study left (right) [Formula: see text]-semihereditary rings over any associative ring, and these rings are exactly [Formula: see text]-semihereditary rings defined by Mahdou and Tamekkante provided that [Formula: see text] is a commutative ring. Some new characterizations of left [Formula: see text]-semihereditary rings are given. Applications go in three directions. The first is to give a sufficient condition when a finitely presented right [Formula: see text]-module is Gorenstein flat if and only if it is Gorenstein projective provided that [Formula: see text] is left coherent. The second is to investigate the relationships between Gorenstein flat modules and direct limits of finitely presented Gorenstein projective modules. The third is to obtain some new characterizations of semihereditary rings, [Formula: see text]-[Formula: see text] rings and [Formula: see text] rings.

Hereditary and semihereditary rings without nilpotent elements

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.

A necessary and sufficient condition is given for a semilocal domain to have the property that every finitely generated module is a direct sum of cyclic modules. Recent work by Tom Shores and Roger Wiegand [SW] and [SWB] has shed new light on the old question: What is the class of commutative rings for which every finitely generated module is a direct sum of cyclic submodules? Following their terminology, such a ring is called an FGC-ring. In this paper, we are able to settle one of their conjectures [SWB, p. 1279], thus giving a characterization of reduced FGC-rings with Noetherian maximal ideal spectrum. We show that, for R a reduced ring with Noetherian maximal ideal spectrum, R is an FGC-ring and only R is a finite direct sum of h-local Bezout domains and each localization of R is an almost maximal valuation ring. The if part of the statement above was shown in [SW] to be an easy consequence of results due to Eben Matlis [M]. Other known results on FGC-rings are reviewed in [SW]; the reader is also referred to [SW] for definitions of the concepts above. The examples constructed in [W] show that our main result answers the FGC question for a large class of rings where it was unknown. Of course it would be nice to give a complete characterization of reduced FGC-rings. This would require settling another question in [SW]: Does every FGC-ring have Noetherian maximal ideal spectrum? They show the answer is yes the ring has fewer than 2' prime ideals [SW, Corollary 6.8]. 1. The main theorem. The result stated in the introduction will easily follow from the Theorem below: Theorem. Let R be an FGC-domain. Then every nonzero prime ideal P is contained in a unique maximal ideal. Proof. Suppose R contains two distinct maximal ideals M and N with Received by the editors May 16, 1974. AMS (MOS) subject classifications (1970). Primary 13C05, 13F05; Secondary 13A15. Copyright ? 1975, Anierican Mathematical Society

The Canonical Element Conjecture holds for those 3 3 -dimensional local rings of depth 2 whose second local cohomology module is a finite direct sum of cyclic modules, and for those 3 3 -dimensional local rings whose second local cohomology module is cyclic.

The fundamental theorem of abelian groups states that any finitely generated abelian group is a direct sum of cyclic groups. This theorem plays a fundamental role in the structure theory of abelian groups. It has fascinated many algebraists to look at this theorem from different points of view:- (a) To find suitable generalizations of this theorem for modules over certain classes of rings, e.g. Dedekind domain, hereditary noetherian prime rings, valuation rings etc. (b) To create suitable versions of this theorem in some module categories and use such versions to develop the structure theory of such module categories. For example, torsion abelian group-like modules, modules with finitely generated submodules direct sums of multiplication modules. (c) To find those rings for which certain versions of the fundamental theorem of abelian groups hold. For example rings over which all finitely generated modules are direct sums of cyclic modules. (d) To examine the structure of certain classes of abelian groups and to try to find modules with similar structures. (e) To find roles of answers to some of above types of questions in the general theory of modules.

<abstract><p>Let $ N $ be a left $ R $-module with the endomorphism ring $ S = \text{End}(_{R}N) $. Given two cardinal numbers $ \alpha $ and $ \beta $ and a matrix $ A\in S^{\beta\times\alpha} $, $ N $ is called flat relative to $ A $ in case, for each $ x\in l_{N^{(\beta)}}(A) = \{u\in N^{(\beta)} \mid uA = 0\} $, there are a positive integer $ k $, $ y\in N^{k} $ and a $ k\times \beta $ row-finite matrix $ C $ over $ S $ such that $ CA = 0 $ and $ x = yC $. It is shown that $ N_{S} $ is flat relative to a matrix $ A $ if and only if $ l_{N^{(\beta)}}(A) $ is generated by $ N $. $ S $ is called left coherent relative to $ A $ if Ker$ (_{S}S^{(\beta)}\to _{S}S^{(\beta)}A) $ is finitely generated. It is shown that $ S $ is left coherent relative to $ A $ if and only if Hom$ _{R}(N, l_{N^{n}}(A)) $ is a finitely generated left $ S $-module if and only if $ l_{N^{n}}(A) $ has an add$ (N) $-precover (add$ (N) $ denotes the category of all direct summands of finite direct sums of copies of $ _{R}N $). Regarding applications, new necessary and sufficient conditions for epic (monic, having the unique mapping property) add$ (N) $-precovers of $ l_{N^{(\beta)}}(A) $ are investigated. Also, some new characterizations of left $ n $-semihereditary rings and von Neumann regular rings are given.</p></abstract>

The direct sum behaviour of its projective modules is a fundamental property of any ring. Hereditary Noetherian prime rings are perhaps the only noncommutative Noetherian rings for which this direct sum behaviour (for both finitely and infinitely generated projective modules) is well-understood, yet highly nontrivial . This book surveys material previously available only in the research literature. It provides a re-worked and simplified account, with improved clarity, fresh insights and many original results about finite length modules, injective modules and projective modules. It culminates in the authors' surprisingly complete structure theorem for projective modules which involves two independent additive invariants: genus and Steinitz class. Several applications demonstrate its utility. The theory, extending the well-known module theory of commutative Dedekind domains and of hereditary orders, develops via a detailed study of simple modules. This relies upon the substantial account of idealiser subrings which forms the first part of the book and provides a useful general construction tool for interesting examples. The book assumes some knowledge of noncommutative Noetherian rings, including Goldie's theorem. Beyond that, it is largely self-contained, thanks to the appendix which provides succinct accounts of Artinian serial rings and, for arbitrary rings, results about lifting direct sum decompositions from finite length images of projective modules. The appendix also describes some open problems.

All rings are associative with identity element 1 and all modules are unital. A ring has enough invertible ideals if every ideal containing a regular element contains an invertible ideal. Lenagan [8, Theorem 3.3] has shown that right bounded hereditary Noetherian prime rings have enough invertible ideals. The proof is quite ingenious and involves the theory of cycles developed by Eisenbud and Robson in [5] and a theorem which shows that any ring S such that R ⊆ S ⊆ Q satisfies the right restricted minimum condition, where Q is the classical quotient ring of R. In Section 1 we give an elementary proof of Lenagan's theorem based on another result of Eisenbud and Robson, namely every ideal of a hereditary Noetherian prime ring can be expressed as the product of an invertible ideal and an eventually idempotent ideal (see [5, Theorem 4.2]). We also take the opportunity to weaken the conditions on the ring R.

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.

By using the transformationsSA(x)R(x), whereS andR(x) are invertible matrices, we reduce a polynomial matrixA(x) whose elementary divisors are pairwise relatively prime to a direct sum of irreducible triangular summands with invariant factors on the principal diagonals.

Let F be an algebraically closed field of characteristic not 2. Define the orthogonal group, O n(F), as the subgroup of GL n(F) consisting of matrices X such that X-1 = X', the transpose of X. We prove that any A ∈ O n(F) having no elementary divisors (t ± 1)k, with k even, is conjugate to a direct sum of zigzag matrices. A zigzag matrix is a special kind of a band matrix with only five nonzero diagonals. We also propose a conjecture about the case excluded above.

In this paper we prove a new characterisation of hereditary PI rings, namely we show that a Noetherian, but not Artinian, PI ringR that is an order in an Artinian ring splits into a direct sum of an Artinian ring of finite representation type and hereditary semiprime rings if and only if all its proper Artinian factor rings are of finite representation type. We also show, through examples, that the above characterisation does not hold for some more general settings.