the author studies the $bf r$$g$-module $a$ such that $bf r$ is an associative ring‎, ‎a group $g$ has infinite section $p$-rank (or infinite 0-rank)‎, ‎$c_{g}(a)=1$‎, ‎and for every‎ ‎proper subgroup $h$ of infinite section $p$-rank (or infinite 0-rank respectively) the quotient module $a/c_{a}(h)$ is‎ ‎a finite $bf r$-module‎. ‎it is proved that if the group $g$ under‎ ‎consideration is local...

A differential ring of analytic functions in several complex variables is called a ring of Noetherian functions if it is finitely generated as a ring and contains the ring of all polynomials. In this paper, we give an effective bound on the multiplicity of an isolated solution of a system of n equations fi = 0 where fi belong to a ring of Noetherian functions in n complex variables. In the one-...

Let $R$ be a commutative Noetherian ring with non-zero identity, $\mathfrak{a}$ an ideal of $R$, $M$ finitely generated $R$--module, and $X$ arbitrary $R$--module. In this paper, for non-negative integers $s, t$ $R$--module $N$, we study the membership $\operatorname{Ext}_{R}^{s}(N, \operatorname{H}^{t}_{\mathfrak{a}}(M, X))$ in Serre subcategories category $R$--modules present some upper bound...

Let R = ⊕ n≥0 Rn be a Noetherian homogeneous ring with onedimensional local base ring (R0,m0). Let q0 ⊆ R0 be an m0-primary ideal, let M be a finitely generated graded R-module and let i ∈ N0. Let Hi R+ (M) denote the i-th local cohomology module of M with respect to the irrelevant ideal R+ := ⊕ n>0 Rn of R. We show that the first Hilbert-Samuel coefficient e1 ( q0, Hi R+ (M)n ) of the n-th gra...

Let K be a field, and let R = K[X] be the polynomial ring in an infinite collection X of indeterminates over K. Let SX be the symmetric group of X. The group SX acts naturally on R, and this in turn gives R the structure of a left module over the group ring R[SX ]. A recent theorem of Aschenbrenner and Hillar states that the module R is Noetherian. We address whether submodules of R can have an...

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...

A Noetherian integral domain R is said to be a splinter if it is a direct summand, as an R–module, of every module–finite extension ring, see [Ma]. In the case that R contains the field of rational numbers, it is easily seen that R is splinter if and only if it is a normal ring, but the notion is more subtle for rings of characteristic p > 0. It is known that F–regular rings of characteristic p...

The notion of the Cousin complex of a module was given by Sharp in 1969. It wasn’t known whether its cohomologies are finitely generated until recently. In 2001, Dibaei and Tousi showed that the Cousin cohomologies of a finitely generated A-module M are finitely generated if the base ring A is local, has a dualizing complex, M satisfies Serre’s (S2)-condition and is equidimensional. In the pres...

The purpose of this note is to point out an interesting connection between the structure of a commutative, Noetherian ring of prime characteristic as a (left) module over its ring of diierential operators and various well studied properties such as F-purity, F-regularity, and strong F-regularity. Theorem 2.2 establishes the rst connections between the D-module structure of rings of characterist...

Given an ideal a in A[x1, . . . , xn] where A is a Noetherian integral domain, we propose an approach to compute the Krull dimension of A[x1, . . . , xn]/a, when the residue class ring is a free A-module. When A is a field, the Krull dimension of A[x1, . . . , xn]/a has several equivalent algorithmic definitions by which it can be computed. But this is not true in the case of arbitrary Noetheri...