we consider the class $mathfrak m$ of $bf r$--modules where $bf r$ is an associative ring. let $a$ be a module over a group ring $bf r$$g$, $g$ be a group and let $mathfrak l(g)$ be the set of all proper subgroups of $g$. we suppose that if $h in mathfrak l(g)$ then $a/c_{a}(h)$ belongs to $mathfrak m$. we investigate an $bf r$$g$--module $a$ such that $g not = g'$, $c_{g}(a) = 1$. we stud...

1. Let R be a ring with unity. An R-module M is said to be balanced or to have the double centralizer property, if the natural homomorphism from R to the double centralizer of M is surjective. If all left and right K-modules are balanced, R is called balanced. It is well known that every artinian uniserial ring is balanced. In [5], J. P. Jans conjectured that those were the only (artinian) bala...

We apply set-theoretic methods to study projective modules and their generalizations over transfinite extensions of simple artinian rings R. prove that if R is small, then the Weak Diamond implies projectivity an arbitrary module can be tested at layer epimorphisms

Let S be a finitely generated standard multigraded algebra over an Artinian local ring A; M a finitely generated multigraded S-module. This paper answers to the question when mixed multiplicities of M are positive and characterizes them in terms of lengths of A-modules. As an application, we get interesting results on mixed multiplicities of ideals, and recover some early results in [Te] and [TV].

If M is a simple module over a ring R then, by the Schur’s lemma, the endomorphism ring of M is a division ring. However, the converse of this result does not hold in general, even when R is artinian. In this short note, we consider perfect rings for which the converse assertion is true, and we show that these rings are exactly the primary decomposable ones.

For an Artinian (n− 1)-Auslander algebra Λ with global dimension n(≥ 2), we show that if Λ admits a trivial maximal (n − 1)-orthogonal subcategory of modΛ, then Λ is a Nakayama algebra and the projective or injective dimension of any indecomposable module in modΛ is at most n− 1. As a result, for an Artinian Auslander algebra with global dimension 2, if Λ admits a trivial maximal 1-orthogonal s...

Let Λ be an Auslander’s 1-Gorenstein Artinian algebra with global dimension 2. If Λ admits a trivial maximal 1-orthogonal subcategory of modΛ, then for any indecomposable module M ∈ modΛ, we have that the projective dimension of M is equal to 1 if and only if so is its injective dimension and that M is injective if the projective dimension of M is equal to 2, which implies that Λ is almost here...

The notion of simple-direct-injective modules which are a generalization injective unifies $C2$ and $C3$-modules. In the present paper, we introduce semisimple-direct-injective module gives unified viewpoint $C2$, $C3$, SSP properties modules. It is proved that ring $R$ Artinian serial with Jacobson radical square zero if only every right $R$-module has and, for any family simple $R$-modules $\...