Let R be a commutative noetherian ring and Γ a finite group. In this paper,we study Gorenstein homological dimensions of modules with respect to a semi-dualizing module over the group ring  . It is shown that Gorenstein homological dimensions  of an  -RΓ module M with respect to a semi-dualizing module,  are equal over R and RΓ  .

Let X be a compact Hausdorff space, and let {Ax : x ∈ X} {Bx collections of Banach algebras such that each Ax is Bx-bimodule. Using the theory bundles spaces as tool, we investigate module amenability certain Ax-valued functions on over Bx-valued X.

If R is a valuation domain of maximal ideal P with a maximal immediate extension of finite rank it is proven that there exists a finite sequence of prime ideals P = L0 ⊃ L1 ⊃ · · · ⊃ Lm ⊇ 0 such that RLj /Lj+1 is almost maximal for each j, 0 ≤ j ≤ m − 1 and RLm is maximal if Lm 6= 0. Then we suppose that there is an integer n ≥ 1 such that each torsion-free R-module of finite rank is a direct s...

Let $(R,underline{m})$ be a commutative Noetherian local ring and $M$ be a non-zero finitely generated $R$-module. We show that if $R$ is almost Cohen-Macaulay and $M$ is perfect with finite projective dimension, then $M$ is an almost Cohen-Macaulay module. Also, we give some necessary and sufficient condition on $M$ to be an almost Cohen-Macaulay module, by using $Ext$ functors.

‎In this paper‎, ‎we introduce the notion of strongly $k-$spaces (with the weak (=finest) pre-topology generated by their strongly compact subsets)‎. ‎We characterize the strongly $k-$spaces and investigate the relationships between preclosedness‎, ‎locally strongly compactness‎, ‎pre-first countableness and being strongly $k-$space.

In this work, we introduce $H^*$-condition on the set of submodules of a module. Let $M$ be a module. We say $M$ satisfies $H^*$ provided that for every submodule $N$ of $M$, there is a direct summand$D$ of $M$ such that $(N+D)/N$ and $(N+D)/D$ are cosingular. We show that over a right perfect right $GV$-ring,a homomorphic image of a $H^*$ duo module satisfies $H^*$.

Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. An $R$-module $M$ is called a multiplication module if for every submodule $N$ of $M$ there exists an ideal $I$ of $R$ such that $N = IM$. It is shown that over a Noetherian domain $R$ with dim$(R)leq 1$, multiplication modules are precisely cyclic or isomorphic to an invertible ideal of $R$. Moreover, we give a charac...

Let G be a locally compact non-compact group. We show that under a very mild assumption on the weight function w, the weighted group algebra L1(G, w) is strongly Arens irregular in the sense of [Dal–Lam–Lau 01]. To this end, we first derive a general factorization theorem for bounded families in the L∞ ( G, w ) ∗ -module L∞ ( G, w ) .

Abstract. Let (R,P) be a Noetherian unique factorization do-main (UFD) and M be a finitely generated R-module. Let I(M)be the first nonzero Fitting ideal of M and the order of M, denotedord_R(M), be the largest integer n such that I(M) ⊆ P^n. In thispaper, we show that if M is a module of order one, then either Mis isomorphic with direct sum of a free module and a cyclic moduleor M is isomorphi...