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.

We investigate Buchbaum and Eisenbud's construction of the second symmetric power $s_R(X)$ of a chain complex $X$ of modules over a commutative ring $R$. We state and prove a number of results from the folklore of the subject for which we know of no good direct references. We also provide several explicit computations and examples. We use this construction to prove the following vers...

the concept of fuzzy soft γ-ring is introduced; and some properties of fuzzy soft γ-rings are given. then the definitions of fuzzy soft γ-ideals are proposed and some of their theories are considered.

we investigate buchbaum and eisenbud's construction of the second symmetric power $s_r(x)$ of a chain complex $x$ of modules over a commutative ring $r$. we state and prove a number of results from the folklore of the subject for which we know of no good direct references. we also provide several explicit computations and examples. we use this construction to prove the following ...

Token Ring network managers are faced with the difficult decision of staying with Token Ring or migrating to Ethernet. Making matters worse is that analysts, consultants, and vendors do not always agree on the answer. Some recommend abandonment of Token Ring as soon as possible and move to Ethernet. Others suggest that Token Ring is strategic and one should stay with it forever. Ultimately, one...

Let k be a field, possibly of finite characteristic, V a finite dimensional vector space over k of dimension r, and W ⊂ Autk(V ) a finite subgroup. There is a natural action of W on the k-dual V # of V , as well as on the symmetric algebra S = S(V ). The algebra S is isomorphic to a polynomial algebra over k on r generators, and we are interested in the question of when the invariant algebra R ...