We introduce and study the concept of $alpha $-semi short modules.Using this concept we extend some of the basic results of $alpha $-short modules to $alpha $-semi short modules.We observe that if $M$ is an $alpha $-semi short module then the dual perfect dimension of $M$ is $alpha $ or $alpha +1$.%In particular, if a semiprime ring $R$ is $alpha $-semi short as an $R$-module, then its Noetheri...

ABSTRACT. Let R be a commutative noetherian ring, I and J are two ideals of R. Inthis paper we introduce the concept of (I;J)- minimax R- module, and it is shown thatif M is an (I;J)- minimax R- module and t a non-negative integer such that HiI;J(M) is(I;J)- minimax for all i

Let R be a Noetherian commutative ring with identity, graded by the nonnegative integers N. Thus the additive group of R has a direct-sum decomposition R = R, + R, + ..., where RiRi C R,+j and 1 E R, . I f in addition R, is a field K, so that R is a k-algebra, we will say that R is a G-akebra. The assumption that R is Noetherian implies that a G-algebra is finitely generated (as an algebra over...

The twisted homogeneous coordinate ring is one of the basic constructions of the noncommutative projective geometry of Artin, Van den Bergh, and others. Chan generalized this construction to the multi-homogeneous case, using a concept of right ampleness for a finite collection of invertible sheaves and automorphisms of a projective scheme. From this he derives that certain multi-homogeneous rin...

In this paper we define and explore the analytic spread ℓ ( I ) $\ell (\mathcal {I})$ of a filtration in local ring. We show that, especially for divisorial symbolic filtrations, some basic properties an ideal extend to even when is non-Noetherian. also illustrate significant differences between with examples. case $I$ , have classical bounds ht ⩽ dim R $\mbox{ht}(I)\leqslant \ell (I)\leqslant ...

Recall that a commutative domain R in which every finitely generated ideal is principal is called a Bézout domain. By definition, a noetherian Bézout domain is a principal ideal domain. Several examples of non-noetherian Bézout domains are listed in [1], 243-246. Recall also that a commutative domain R is called an Elementary Divisor domain if, given any matrix A with coefficients in R, there e...

Unlike the Gorenstein projective and injective dimensions, the majority of results on the Gorenstein flat dimension have been established only over Noetherian (or coherent) rings. Naturally, one would like to generalize these results to any associative ring. In this direction, we show that the Gorenstein flat dimension is a refinement of the classical flat dimension over any ring; and we invest...