相关论文: An elementary result on infinite and finite direct…

A direct sum decomposition theory is developed for direct summands (and complements) of modules over a semiring $R$, having the property that $v+w = 0$ implies $v = 0$ and $w = 0$. Although this never occurs when $R$ is a ring, it always…

This paper is devoted to the study of the endo-structure of infinite direct sums $\bigoplus_{i \in I} M_i$ of indecomposable modules $M_i$ over a ring $R$. It is centered on the following question: If $S = \text{End}_R \bigl( \bigoplus_{i…

Hard to summarize concisely; here are the high points. The first two statements below are ring-theoretic; in these R is a nontrivial ring, R^\omega, and \bigoplus_\omega R are the direct product, respectively direct sum, of countably many…

A module $M$ is {called} stable if it has no nonzero projective direct summand. For a ring $ R $, we study conditions under which $R$-modules from certain classes decompose as a direct sum of a projective submodule and a stable submodule.…

Let $Q$ be a Noetherian ring with finite Krull dimension and let $\mathbf{f}= f_1,... f_c$ be a regular sequence in $Q$. Set $A = Q/(\mathbf{f})$. Let $I$ be an ideal in $A$, and let $M$ be a finitely generated $A$-module with $\projdim_Q…

Let R be a ring, M a left R-module, I an infinite set, N either the direct sum or product of |I| copies of M, and E the endomorphism ring of N as a left R-module. In this note it is shown that E is not the union of a chain of |I| or fewer…

Let $R$ be a finite ring and let $M, N$ be two finite left $R$-modules. We present two distinct deterministic algorithms that decide in polynomial time whether or not $M$ and $N$ are isomorphic, and if they are, exhibit an isomorphism. As…

In this paper we show that for a given set of pairwise comaximal ideals $\{X_i\}_{i\in I}$ in a ring $R$ with unity and any right $R$-module $M$ with generating set $Y$ and $C(X_i)=\sum\limits_{k\in\mathbb{N}}\underline{\ell}_M(X_i^{k})$,…

Let $\Lambda$ be an artin algebra. We are going to consider full subcategories of $\mod\Lambda$ closed under finite direct sums and under submodules with infinitely many isomorphism classes of indecomposable modules. The main result asserts…

A module over a ring $R$ is pure projective provided it is isomorphic to a direct summand of a direct sum of finitely presented modules. We develop tools for the classification of pure projective modules over commutative noetherian rings.…

Monogeny classes and epigeny classes have proved to be useful in the study of direct sums of uniserial modules and other classes of modules. In this paper, we show that they also turn out to be useful in the study of direct products.

Let $\mathcal C$ be a class of modules over a ring $R$, closed under direct sums over index sets of cardinality $\kappa$ and isomorphisms, and such that the isomorphism classes form a set. The monoid of modules $V(\mathcal C)$ encodes the…

It is shown that, if $R$ is either an Artin algebra or a commutative noetherian domain of Krull dimension $1$, then infinite direct products of $R$-modules resist direct sum decomposition as follows: If $(M_n)_{n \in \Bbb N}$ is a family of…

We obtain an asymptotic upper bound for the smallest number of generators for a finite direct sum of matrix algebras with entries in a finite field. This produces an upper bound for a similar quantity for integer matrix rings. We also…

We show that every finitely generated cohomologically trivial module over $RG$, where $G$ is a finite $p$-group and $R$ is a $p$-adic ring, splits as the direct sum of a finite cohomologically trivial $RG$-module and a free $RG$-module.…

We show that every unconditional basis in a finite direct sum $\bigoplus_{p\in A} \ell_p$, with $A\subset (0,\infty]$, splits into unconditional bases of each summand. This settles a 40 year old question raised in [A. Orty\'nski,…

We prove that the module categories of Noether algebras (i.e., algebras module finite over a noetherian center) and affine noetherian PI algebras over a field enjoy the following product property: Whenever a direct product $\prod_{n \in…

In this paper, we study the class of modules have the property that every pure submodule is essential in a direct summand. These modules are termed as pure extending modules which is a proper generalisation of extending modules. Examples…

For each group G which decomposes into a finitary direct product of free groups of finite rank we construct a regular band B such that the free idempotent generated semigroup over B contains a maximal subgroup isomorphic to G. In…

We introduce a similarity relation between submodules of a module $M$ over a ring $R$, extending the classical notion of similarity for right ideals. Focusing on (faithfully) projective modules, we establish a sharp lower bound for the…