Related papers: Radical-injectivy in the category S-Act

In abelian categories like the category of R-modules and even in the category S-Act 0 of S-acts with a unique zero, idempotent radicals and torsion theories are equivalent, and the {\tau}-torsion and {\tau}- torsion free classes of a…

In this article we give application of closure operators in category of modules. Our main result shows that every subcategory A of injective modules of R-mod (under a mild condition) induces a torsion theory of R-mod.

Suppose that $(\mathcal{F},\mathcal{M})$ is an injective structure of $R$-Mod such that the class $\mathcal{F}$ is closed for direct limits, then two modules in $\mathcal{M}$ are isomorphic if there are maps in $\mathcal{F}$ from each one…

We introduce and investigate ss-injectivity as a generalization of both soc-injectivity and small injectivity. A module M is said to be ss-N-injective (where N is a module) if every R-homomorphism from a semisimple small submodule of N into…

Let $R$ a commutative ring, $\mathfrak{a} \subset R$ an ideal, $I$ an injective $R$-module and $S \subset R$ a multiplicatively closed set. When $R$ is Noetherian it is well-known that the $\mathfrak{a}$-torsion sub-module…

Let $R$ be a Noetherian ring and let $C$ be a semidualizing $R$-module. In this paper, we are concerned with the tensor and torsion product of $C$-injective modules. Firstly, it is shown that the tensor product of any two $C$-injective…

A commutative ring is said to have ITI with respect to an ideal a if the a-torsion functor preserves injectivity of modules. Classes of rings with ITI or without ITI with respect to certain sets of ideals are identified. Behaviour of ITI…

Let R be a commutative ring and S be an R-algebra. It is well-known that if N is an injective R-module, then Hom(S,N) is an injective S-module. The converse is not true, not even if R is a commutative noetherian local ring and S is its…

Let $R$ be a commutative Noetherian ring. We give criteria for flatness of $R$-modules in terms of associated primes and torsion-freeness of certain tensor products. This allows us to develop a criterion for regularity if $R$ has…

Let $R$ be a commutative ring with identity, and let $S$ be a multiplicative subset of $R$. In this paper, we introduce the notion of $S$-injective modules as a weak version of injective modules. Among other results, we provide an…

Let $R$ be a regular domain containing a field $K$ of characteristic zero and let $D$ be the ring of $K$-linear differential operators on $R$. Let $E$ be an injective left $D$-module. We ask the question, when is $E$ injective as a…

Let R be a commutative ring, and let S be a multiplicative subset of R. In this paper, we introduce and investigate the notion of S-FP-injective modules. Among other results, we show that, under certain conditions, a ring R is S-Noetherian…

Let $T_R(M)$ be a tensor ring and $\mathcal{X}$, $\mathcal{Y}$ be two classes of $R$-modules. Under certain conditions, we prove that a $T_R(M)$-module $(A, u)$ is $Ind(\mathcal{X})$-Gorenstein projective if and only if $u$ is monomorphic…

In this article, we give a generalization to injective modules by using $e$-exact sequences introduced by Akray in [1] and name it $e$-injective modules and investigate their properties. We reprove both Baer criterion and comparison theorem…

Let R be a ring (not necessarily commutative). A left R-module is said to be cotorsion if Ext 1 R (G, M) = 0 for any flat R-module G. It is well known that each pure-injective left R-module is cotorsion, but the converse does not hold: for…

We formulate and establish a generalization of Koll\'ar's injectivity theorem for adjoint bundles twisted by suitable multiplier ideal sheaves. As applications, we generalize Koll\'ar's torsion-freeness, Koll\'ar's vanishing theorem, and a…

This paper introduces the notion of uniformly-S-pseudo-injective (u-S-pseudo-injective) modules as a generalization of u-S-injective modules. Let R be a ring and S a multiplicative subset of R. An R-module E is said to be…

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 $C_1$, $C_2$, $C_3$,…

A module is called automorphism-invariant if it is invariant under any automorphism of its injective hull. In [Algebras for which every indecomposable right module is invariant in its injective envelope, Pacific J. Math., vol. 31, no. 3…

Let R be a regular semi-local integral domain containing a field and K be its fraction field. Let mu: G --> T be an R-group schemes morphism between reductive R-group schemes, which is smooth as a scheme morphism. Suppose that T is an…