Related papers: Decompositions of modules lacking zero sums

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.…

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.…

Let $R$ be a ring and let $\mathcal C$ be a small class of right $R$-modules which is closed under finite direct sums, direct summands, and isomorphisms. Let $\mathcal V (\mathcal C)$ denote a set of representatives of isomorphism classes…

Let $R$ be a ring, and consider a left $R$-module given with two (generally infinite) direct sum decompositions, $A\oplus(\bigoplus_{i\in I} C_i)=M=B\oplus(\bigoplus_{j\in J} D_j),$ such that the submodules $A$ and $B$ and the $D_j$ are…

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…

An $R$-module $V$ over a semiring $R$ lacks zero sums (LZS) if $ x +y = 0 \; \Rightarrow \; x = y = 0$. More generally, asubmodule $W$ of $V$ is "summand absorbing", if $ \forall \, x, y \in V: \ x + y \in W \; \Rightarrow \; x \in W, \; y…

Let $R$ be a (possibly noncommutative) ring and let $\mathcal C$ be a class of finitely generated (right) $R$-modules which is closed under finite direct sums, direct summands, and isomorphisms. Then the set $\mathcal V (\mathcal C)$ of…

We study direct sum decompositions of modules satisfying the descending chain condition on direct summands. We call modules satisfying this condition Krull-Schmidt artinian. We prove that all direct sum decompositions of Krull-Schmidt…

A submodule $W$ of $V$ is summand absorbing, if $x + y \in W$ implies $x \in W, \; y \in W $ for any $x, y \in V$. Such submodules often appear in modules over (additively) idempotent semirings, particularly in tropical algebra. This paper…

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…

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…

We describe and prove correctness of two practical algorithms for finding indecomposable summands of finitely generated modules over a finitely generated k-algebra R. The first algorithm applies in the (multi)graded case, which enables the…

We introduce the notion of pure extending modules, a refinement of classical extending modules in which only pure submodules are required to be essential in direct summands. Fundamental properties and characterizations are established,…

We study the structure of certain modules $V$ over linear spaces $W$ with restrictions neither on the dimensions nor on the base field $\mathbb F$. A basis $\mathfrak B = \{v_i\}_{i\in I}$ of $V$ is called multiplicative respect to the…

Categorical constructions on heaps and modules over trusses are considered and contrasted with the corresponding constructions on groups and rings. These include explicit description of free heaps and free Abelian heaps, coproducts or…

It is proved that the sum of the Loewy lengths of the homology modules of a finite free complex F over a local ring R is bounded below by a number depending only on R. This result uncovers, in the structure of modules of finite projective…

This survey article is devoted to the notions of purity, algebraic and $\Sigma$-algebraic compactness, direct sum decompositions, and representation type in the category of modules over a ring. It begins with basic definitions, a brief…

We prove two criteria for direct sum decomposability of homogeneous polynomials. For a homogeneous polynomial with a non-zero discriminant, we interpret direct sum decomposability of the polynomial in terms of factorization properties of…

We say that an $R$-module $M$ is {\it virtually simple} if $M\neq (0)$ and $N\cong M$ for every non-zero submodule $N$ of $M$, and {\it virtually semisimple} if each submodule of $M$ is isomorphic to a direct summand of $M$. We carry out a…

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…