$R$-module. In this paper, we explore more properties of $Max$-injective modules and we study some conditions under which the maximal spectrum of $M$ is a $Max$-spectral space for its Zariski topology.

Let $T=\bigl(\begin{smallmatrix}A&0\U\&B\end{smallmatrix}\bigr)$ be a formal triangular matrix ring, where $A$ and $B$ are rings $U$ is $(B, A)$-bimodule. We prove: (1) If $U\_{A}$ ${B}U$ have finite flat dimensions, then left $T$-module $\bigl(\begin{smallmatrix}M\_1\ M\_2\end{smallmatrix}\bigr){\varphi^{M}}$ Ding projective if only $M\_1$ $M\_2/{\operatorname{im}(\varphi^{M})}$ the morphism $...

Gillespie posed two questions in [Front. Math. China 12 (2017) 97-115], one of which states that “for what rings R do we have K(AC)=K(R-Inj)?”. We give an answer to such a question. As applications, obtain new homological approach unifies some well-known conditions Krause’s recollement holds, and example show there exists Gorenstein injective module is not AC-injective. also improve Neeman’s an...

we show that every semi-artinian module which is contained in a direct sum of finitely presented modules in $si[m]$, is weakly co-semisimple if and only if it is regular in $si[m]$. as a consequence, we observe that every semi-artinian ring is regular in the sense of von neumann if and only if its simple modules are $fp$-injective.

A generalization of injective modules (noted GI-modules), distinct from p-injective modules, is introduced. Rings whose p-injective modules are GI are characterized. If M is a left GI-module, E = End(AM), then E/J(E) is von Neumann regular, where J(E) is the Jacobson radical of the ring E. A is semisimple Artinian if, and only if, every left A-module is GI. If A is a left p. p., left GI-ring su...

It is proved that localizations of injective R-modules of finite Goldie dimension are injective if R is an arithmetical ring satisfying the following condition: for every maximal ideal P , RP is either coherent or not semicoherent. If, in addition, each finitely generated R-module has finite Goldie dimension, then localizations of finitely injective R-modules are finitely injective too. Moreove...

An R-module M is called epi-retractable if every submodule of MR is a homomorphic image of M. It is shown that if R is a right perfect ring, then every projective slightly compressible module MR is epi-retractable. If R is a Noetherian ring, then every epi-retractable right R-module has direct sum of uniform submodules. If endomorphism ring of a module MR is von-Neumann regular, then M is semi-...

The concepts of free modules, projective modules, injective modules, and the like form an important area in module theory. The notion of free fuzzy modules was introduced by Muganda as an extension of free modules in the fuzzy context. Zahedi and Ameri introduced the concept of projective and injective L-modules. In this paper, we give an alternate definition for injective L-modules and prove t...

in this paper‎, ‎we introduce the notion of $(m,n)$-‎algebr‎aically compact modules as an analogue of algebraically‎ ‎compact modules and then we show that $(m,n)$-algebraically‎ ‎compactness‎ ‎and $(m,n)$-pure injectivity for modules coincide‎. ‎moreover‎, ‎further characterizations of a‎ ‎$(m,n)$-pure injective module over a commutative ring are given‎.

For a module-finite algebra over commutative noetherian ring, we give complete description of flat cotorsion modules in terms prime ideals the algebra, as generalization Enochs' result for ring. As consequence, show that pointwise Matlis duality gives bijective correspondence between isoclasses indecomposable injective left and right modules. This is an explicit realization Herzog's homeomorphi...