It is shown that every commutative local ring of bounded module type is an almost maximal valuation ring. We say that an associative commutative unitary ring R is of bounded module type if there exists a positive integer n such that every finitely generated R-module is a direct sum of submodules generated by at most n elements. For instance, every Dedekind domain has bounded module type with bo...

Nakayama (Ann. of Math. 42, 1941) showed that over an artinian serial ring every module is a direct sum of uniserial modules. Hence artinian serial rings have the property that each right (left) ideal is a finite direct sum of quasi-injective right (left) ideals. A ring with the property that each right (left) ideal is a finite direct sum of quasi-injective right (left) ideals will be called a ...

A generalization of injective modules (noted GI-modules), distinct from p-injective modules, is introduced. Rings whose p-injective modules are GI are characterized. If M is a left GI-module, E = End(AM), then E/J(E) is von Neumann regular, where J(E) is the Jacobson radical of the ring E. A is semisimple Artinian if, and only if, every left A-module is GI. If A is a left p. p., left GI-ring su...

We prove versions of results of Foxby and Holm about modules of finite (Gorenstein) injective dimension and finite (Gorenstein) projective dimension with respect to a semidualizing module. We also verify two special cases of a question of Takahashi and White.

In the context of comparative analysis of protein-protein interaction graphs, we use a graph-based formalism to detect the preservation of a given protein complex G in the protein-protein interaction graphH of another species with respect to (w.r.t.) orthologous proteins. Two problems are considered: the Exact-(μG, μH)-Matching problem and the Max-(μG, μH)-Matching problems, where μG (resp. μH)...

Let R be a ring. A right R-module is said to C-flat if the kernel of any epimorphism B → C-pure in B, i.e. induced map Hom(C,B) Hom(C,A) surjective for cyclic C. Projective modules are and weakly-flat neat-flat. In this article, it discussed connections between C-flat, weakly-flat, neat-flat singly flat modules. It shown that coincide with singly-projective over arbitrary rings. Next, several c...

Let $R$ be a ring, $n$ an non-negative integer and $d$ positive or $\infty$. A right $R$-module $M$ is called \emph{$(n,d)^*$-projective} if ${\rm Ext}^1_R(M, C)=0$ for every $n$-copresented $C$ of injective dimension $\leq d$; ring \emph{right $(n,d)$-cocoherent} with $id(C)\leq d$ $(n+1)$-copresented; $(n,d)$-cosemihereditary} whenever $0\rightarrow C\rightarrow E\rightarrow A\rightarrow 0$ e...

Let (R,m) be a commutative Noetherian local ring. We establish some bounds for the sequence of Bass numbers and their dual for a finitely generated R-module. Introduction Throughout this paper, (R,m, k) is a non-trivial commutative Noetherian local ring with unique maximal ideal m and residue field k. Several authors have obtained results on the growth of the sequence of Betti numbers {βn(k)} (...

For a one-dimensional local domain CM constructed by Akizuki, we find residue maps which give rise to a local duality. The completion of CM is described using these residue maps. Injective hulls of a given module are all isomorphic. For this reason, people often speak of the injective hull to indicate its “uniqueness”. However, isomorphisms between these injective hulls are not canonical. In fa...

We study certain filtrations of indecomposable injective modules over classical Lie superalgebras, applying a general approach for noetherian rings developed by Brown, Jategaonkar, Lenagan, and Warfield. To indicate the consequences of our analysis, suppose that g is a complex classical simple Lie superalgebra and that E is an indecomposable injective g-module with nonzero (and so necessarily s...