Let R be a ring and let M be a right R-module with S End MR . M is called almost general quasiprincipally injective or AGQP-injective for short if, for any 0/ s ∈ S, there exist a positive integer n and a left ideal Xsn of S such that s / 0 and lS Ker s Ss ⊕ Xsn . Some characterizations and properties of AGQP-injective modules are given, and some properties of AGQP-injective modules with additi...

A module is called uniseriat if it has a unique composition series of finite length. A ring (always with 1) is called serial if its right and left free modules are direct sums of uniserial modules. Nakayama, who called these rings generalized uniserial rings, proved [21, Theorem 171 that every finitely generated module over a serial ring is a direct sum of uniserial modules. In section one we g...

Relative to a hereditary torsion theory $tau$ we introduce a dimension for a module $M$, called {em $tau$-rank of} $M$, which coincides with the reduced rank of $M$ whenever $tau$ is the Goldie torsion theory. It is shown that the $tau$-rank of $M$ is measured by the length of certain decompositions of the $tau$-injective hull of $M$. Moreover, some relations between the $tau$-rank of $M$ and c...

relative to a hereditary torsion theory $tau$ we introduce a dimension for a module $m$, called {em $tau$-rank of} $m$, which coincides with the reduced rank of $m$ whenever $tau$ is the goldie torsion theory. it is shown that the $tau$-rank of $m$ is measured by the length of certain decompositions of the $tau$-injective hull of $m$. moreover, some relations between the $tau$-rank of $m$ and c...

In [Hov02], the second author introduced the Gorenstein projective and Gorenstein injective model structures on R-Mod, the category of R-modules, where R is any Gorenstein ring. These two model structures are Quillen equivalent and in fact there is a third equivalent structure we introduce; the Gorenstein flat model structure. The homotopy category with respect to each of these is called the st...