Related papers: Modules close to SSP- and SIP-modules

It is well-known that a ring R is semiperfect if and only if R as a left (or as a right) R-module is a supplemented module. Considering weak supplements instead of supplements we show that weakly supplemented modules M are semilocal (i.e.,…

In this article, we study the notion of a finite $\Sigma$-Rickart module, as a module theoretic analogue of a right semi-hereditary ring. A module $M$ is called \emph{finite $\Sigma$-Rickart} if every finite direct sum of copies of $M$ is a…

Injective, pure-injective and fp-injective modules are well known to provide for approximations in the category Mod-R for an arbitrary ring R. We prove that this fails for many other generalizations of injectivity: the $C_1$, $C_2$, $C_3$,…

We give a simplified proof of Tits' classification of semisimple algebraic groups that remains valid over semilocal rings. In particular, we provide explicit necessary and sufficient conditions that anisotropic groups of a given type appear…

In a recent paper of Alahmadi, Alkan and Lopez-Permouth, a ring R is defined to have no (simple) middle class if the injectivity domain of any (simple) R-module is the smallest or largest possible. Er, Lopez-Permouth and Sokmez use this…

Let $G$ be a connected reductive algebraic group defined over an algebraically closed field %$k$ of characteristic $p > 0$. Our first aim in this note is to give concise and uniform proofs for two fundamental and deep results in the context…

In this short note, we construct a right adjoint to the functor which associates to a ring $R$ equipped with a group action its twisted group ring. This right adjoint admits an interpretation as semilinearization, in that it sends an…

The purpose of this paper and its sequel, is to introduce a new class of modules over a commutative ring $R$, called $\mathbb{P}$-radical modules (modules $M$ satisfying the prime radical condition "$(\sqrt[p]{{\cal{P}}M}:M)={\cal{P}}$" for…

In this article, we introduce the notion of uniformly S-projective (u-S-projective) relative to a module. Let S be a multiplicative subset of a ring R and M an R-module. An R-module P is said to be u-S-projective relative to M if for any…

Among the finitely generated modules over a Noetherian ring R, the semidualizing modules have been singled out due to their particularly nice duality properties. When R is a normal domain, we exhibit a natural inclusion of the set of…

Let $A$ be a simple C*-algebra of stable rank one and let $p$ and $q$ be two $\sigma$-compact open projections. It is proved that there is a continuous path of unitaries in ${\tilde A}$ which connects open sub-projections of $p$ which is…

Inspired from the work of P. Scholze on the finiteness of \(\mathbf{F}_{p}\)-cohomology groups of proper rigid-analytic varieties over \(p\)-adic fields, Zavyalov recently introduced the notion of almost coherent rings, which plays a key…

We call a ring R pointwise semicommutative if for any element a in R either l(a) or r(a) is an ideal of R. A class of pointwise semicommutative rings is a strict generalization of semicommutative rings. Since reduced rings are pointwise…

Let R be a commutative ring with identity, S be a multiplicatively closed subset of R, and let M be an R-module. The aim of this paper is to introduce the notion of S-secondary submodules of M as a generalization of secondary submodules of…

Let $R$ be a commutative ring, $\pi$ be a finite group, $R\pi$ be the group ring of $\pi$ over $R$. Theorem 1. If $R$ is a commutative artinian ring and $\pi$ is a finite group. Then the Cartan map $c:K_0(R\pi)\to G_0(R\pi)$ is injective.…

Let $(R,\mathfrak{m})$ be a commutative local noetherian ring. For an artinian $R$-module $M$, we show the equality $$\mathrm{cosupp}_RM=\mathrm{Cosupp}_RM$$ using the semi-discrete linearly compactness of $R$-module…

Let $R$ be a ring and $M$ be a right $R$-module. $M$ is called neat-flat if any short exact sequence of the form $0\to K\to N\to M\to 0$ is neat-exact i.e. any homomorphism from a simple right $R$-module $S$ to $M$ can be lifted to $N$. We…

Let $R$ be a commutative ring with identity, $S$ a multiplicatively closed subset of $R$, and $M$ be an $R$-module. In this paper, we study and investigate some properties of $S$-primary submodules of $M$. Among the other results, it is…

We say that a subring $R_0$ of a ring $R$ is semi-invariant if $R_0$ is the ring of invariants in $R$ under some set of ring endomorphisms of some ring containing $R$. We show that $R_0$ is semi-invariant if and only if there is a ring…

All rings considered are commutative. In this article we introduce and study two notions of modules which are stronger than CS modules, namely weakly IN modules and strongly CS modules. Our main aim is to characterize when a trivial…