相关论文: On a question of Mohamed and M\"uller

Let R be an associative ring with unity and let M be an R-module. We call M (ample) Rad-supplementing if M has a (ample) Rad-supplement in every extension. If M is Rad-supplementing, then every direct summand of M is Rad-supplementing, but…

Let $M$ be a left module over a ring $R$ and $I$ an ideal of $R$. $M$ is called an $I$-supplemented module (finitely $I$-supplemented module) if for every submodule (finitely generated submodule) $X$ of $M$, there is a submodule $Y$ of $M$…

It is well known that a direct sum of CLS-modules is not, in general, a CLS-module. It is proved that if $M=M_1\oplus M_2$, where $M_1$ and $M_2$ are CLS-modules such that $M_1$ and $M_2$ are relatively ojective (or $M_1$ is…

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.…

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…

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…

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 $T$ be a subset of a ring $A$, and let $M$ be an $A$-module. We study the additive subgroups $F$ of $M$ such that, for all $x \in M$, if $tx \in F$ for some $t \in T$, then $x \in F$. We call any such subset $F$ of $M$ a $T$-factroid of…

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…

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 $A,B\in B(H)$. In the present paper, we establish simple and interesting facts on when we have $|A||B|=|B||A|$, $|AB|=|A||B|$, $|A\pm B|\leq |A|+|B|$, $||A|-|B||\leq |A\pm B|$ and $\||A|-|B|\|\leq \|A\pm B\|$, where $|\cdot|$ denotes…

A module is said to be \textsf{distributive} if the lattice of its submodules is distributive. A direct sum of distributive modules is called a \textsf{semidistributive} module. In this paper we consider rings $A$ such that all right…

The direct summand conjecture asserts that if R is a regular local ring and S is a module-finite R-algebra containing R, then R is a direct summand of S as an R-module. It was previously known to be true if R contains a field or if dim R is…

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…

We consider modules E over a C*-algebra A which are equipped with a map into A_+ that has the formal properties of a norm. We completely determine the structure of these modules. In particular, we show that if A has no nonzero commutative…

The parallel sum for adjoinable operators on Hilbert $C^*$-modules is introduced and studied. Some results known for matrices and bounded linear operators on Hilbert spaces are generalized to the case of adjointable operators on Hilbert…

Let $G$ be an additive abelian group and $h$ be a positive integer. For a nonempty finite subset $A=\{a_0, a_1,\ldots, a_{k-1}\}$ of $G$, we let \[h_{\underline{+}}A:=\{\Sigma_{i=0}^{k-1}\lambda_{i} a_{i}: (\lambda_{0}, \ldots,…

A cohomological support, Supp_A(M), is defined for finitely generated modules M over an left noetherian ring R, with respect to a ring A of central cohomology operations on the derived category of R-modules. It is proved that if the…

A module M is called principally Goldie*-lifting if for every proper cyclic submodule X of M, there is a direct summand D of M such that $X\beta^*D$. In this paper, we focus on principally Goldie*-lifting modules as generalizations of…

Summand absorbing submodules are common in modules over (additively) idempotent semirings, for example, in tropical algebra. A submodule $W$ of $V$ is summand absorbing, if $x + y \in W$ implies $x \in W, \; y \in W $ for any $x, y \in V$.…