Let R be a ring and M a right R-module. M is called n-FP-projective if Ext M N = 0 for any right R-module N of FP-injective dimension ≤n, where n is a nonnegative integer or n = . R M is defined as sup n M is n-FP-projective and R M = −1 if Ext M N = 0 for some FP-injective right R-module N. The right -dimension r -dim R of R is defined to be the least nonnegative integer n such that R M ≥ n im...

We show that the category of projective modules over a graded commutative ring admits a triangulation with respect to module suspension if and only if the ring is a finite product of graded fields and exterior algebras on one generator over a graded field (with a unit in the appropriate degree). We also classify the ungraded commutative rings for which the category of projective modules admits ...

We show  that a projective maximal submodule of afinitely generated, regular, extending module is a directsummand. Hence, every finitely generated, regular, extendingmodule with projective maximal submodules is semisimple. As aconsequence, we observe that every regular, hereditary, extendingmodule is semisimple. This generalizes and simplifies a result of  Dung and   Smith. As another consequen...

Let G be a finite group, and let Ω := {t ∈ G | t = 1}. Then Ω is a G-set under conjugation. Let k be an algebraically closed field of characteristic 2. It is shown that each projective indecomposable summand of the G-permutation module kΩ is irreducible and self-dual, whence it belongs to a real 2-block of defect zero. This, together with the fact that each irreducible kG-module that belongs to...

Let R be a ring and n a fixed positive integer, we investigate the properties of n-strongly Gorenstein projective, injective and flat modules. Using the homological theory , we prove that the tensor product of an n-strongly Gorenstein projective (flat) right R -module and projective (flat) left R-module is also n-strongly Gorenstein projective (flat). Let R be a coherent ring ,we prove that the...

Our dual notions “locally injective” and “locally projective” modules in Mod-R are good tools to study the relations between the singular, respectively cosingular, submodule of Hom R(M, W ) and the total Tot (M, W ). These notions have further interesting properties.