Let R be a local ring with maximal ideal m such that there exists a pair of elements a, b with (0 : a) = bR and (0 : b) = aR; we say that a pair a, b as above is an exact pair of zero divisors. We study minimal free resolutions of finitely generated R-modules M , with particular attention to the case when m = 0. Let e denote the minimal number of generators of m. If R is Gorenstein with m = 0 a...

Let (R, m, K) be a Gorenstein local ring of prime characteristic p > 0, and let x1, . . . , xn be a system of parameters. Let It = (x1, . . . , x t n)R for all t ≥ 1, and let u ∈ R represent a socle generator in R/I, where I = I1 = (x1, . . . , xn)R. Let y = x1 · · · xn. We have seen that E = lim −→ t R/It is an injective hull of K = R/m over R, where the map R/It → R/It+1 is induced by multipl...

Given a module M over a ring R that has a grading by a semigroup Q, we present a spectral sequence that computes the local cohomology H I(M) at any graded ideal I in terms of Ext modules. We use this method to obtain finiteness results for the local cohomology of graded modules over semigroup rings. In particular we prove that for a semigroup Q whose saturation Q is simplicial, and a finitely g...

Abstract We define the notion of a characteristically inert socle-regular Abelian p -group and explore such groups by focussing on their socles, thereby relating them to previously studied notions socle-regularity. show that large classes -groups, including all divisible, totally projective torsion-complete share this property when prime is odd. The present work generalizes full inertia intensi...

Let $G$ be an automorphism group of a‎ ‎$2$-$(v,k,4)$ symmetric design $mathcal D$‎. ‎In this paper‎, ‎we‎ ‎prove that if $G$ is flag-transitive point-primitive‎, ‎then the‎ ‎socle of $G$ cannot be an exceptional group of Lie type‎.

the main aim of this article is to study (v,k,λ)-symmetric designs admitting a flag-transitive and point-primitive automorphism group g whose socle is psl(3,q). we indeed show that the only possible design satisfying these conditions is a desarguesian projective plane pg(2,q) and g > psl(3,q).