Narrow Results

A module $M$ is called pseudo semi-projective if, for all endomorphisms $\alpha ,\beta$ of $M$ with $\mathrm{\text{Im}}(\alpha )=\mathrm{\text{Im}}(\beta )$, then $\alpha \mathrm{\text{End}}(M)=\beta \mathrm{\text{End}}(M)$. In this paper, we study submodules of pseudo semi-projective modules. It is shown that if every (finitely generated) submodule of a semiprojective right $R$-module is pseudo semi-projective, then every factor ring of $R$ is right (semi-)hereditary. Moreover, we show that if $R$ is left perfect and finitely generated submodules of pseudo semi-projective right $R$-modules are pseudo semi-projective, then $R$ has a decomposition of abelian groups $R_{\mathbb{Z}}=S\oplus J(R)$, where $S$ is a semisimple subring of $R$ containing 1.

In this work, we consider the existence and construction of pseudocomplements in some lattices of module classes. The classes of modules belonging to these lattices are defined via closure under operations such as taking submodules, quotients, extensions, injective hulls, direct sums or products. We characterize the rings for which the lattices $R$-tors (of hereditary torsion classes), $R$-nat (the lattice of natural classes) and $R$-conat (the lattice of conatural classes) coincide.

This paper focusses on the question of when modules have ADS- preenvelopes and covers. For a ring $R$, it is proved that every ADS module over $R$ is injective if and only if every right $R$-module has an ADS-(pre)envelope (an ADS-(pre)cover). In this paper, we also introduce a generalization of ADS modules stated in terms of their invariance under certain automorphisms of their envelopes.

A left $R$-module $M$ is said to be left singly injective if ${\mathrm{\text{Ext}}}_R^1(F/K,M)=0$ for any cyclic submodule $K$ of any finitely generated free left $R$-module $F$. In this paper, we study the notion of singly injective modules which is generalization of injective modules and absolutely pure modules. In this direction, we give conditions which guarantee that each singly injective left $R$-module is either injective or absolutely pure. Finally, we study rings whose simple modules are singly injective (SSI-rings).

It is shown that every finitely generated right $R$-module is almost injective if and only if every cyclic right $R$-module is almost injective, if and only if $R/J(R)$ is a right $SV$-ring with $\mathrm{\text{Loewy}}(R_R)\le 2$ and there is a finite set of orthogonal idempotents ${\{e_i\}}_{i\in I}$ in $R$ such that $e_{i}R$ is an injective local right $R$-module of length two for every $i\in I$ and $J(R)={\oplus }_{i\in I}J(e_{i}R)$.

In this paper, the concepts of essentially slightly compressible modules and essentially slightly compressible rings are introduced, and related properties are investigated. The notion of essentially slightly compressible modules and rings are generalization of essentially compressible modules and rings introduced by [P. F. Smith and M. R. Vedadi, Essentially compressible modules and rings, J. Algebra304 (2006) 812–831]. I have also provided the characterization of such modules in terms of nonsingular injective modules. It has been shown that over a noetherian ring for a uniform module, essentially slightly compressible modules, slightly compressible modules and nonzero homomorphism from M into U for a nonzero uniform submodule U of M are equivalent. Throughout this paper, all rings are associative and all modules are unital right R-modules.