Let R be a left and right Noetherian ring and n, k any non-negative integers. R is said to satisfy the Auslander-type condition Gn(k) if the right flat dimension of the (i+1)-st term in a minimal injective resolution of RR is at most i+ k for any 0 ≤ i ≤ n− 1. In this paper, we prove that R is Gn(k) if and only if so is a lower triangular matrix ring of any degree t over R.

We find a bound for the Goldie dimension of hereditary modules in terms of the cardinality of the generator sets of its quasi-injective hull. Several consequences are deduced. In particular, it is shown that every right hereditary module with countably generated quasi-injective hull is noetherian. Or that every right hereditary ring with finitely generated injective hull is artinian, thus answe...

Given an iterated skew polynomial ring C[y1; τ1, δ1] . . . [yn; τn, δn] over a complete local ring C with maximal ideal m, we prove, under suitable assumptions, that the completion at the ideal m+ 〈y1, y2, . . . , yn〉 is an iterated skew power series ring. When C is a field, this completion is a local, noetherian, Auslander regular domain with Krull, classical Krull and global dimension all equ...

Let $ \pi:X \rightarrow X_{0}$ be a projective morphism of schemes, such that $ X_{0}$ is Noetherian and essentially of finite type over a field $ K$. Let $ i \in \mathbb{N}_{0}$, let $ {\mathcal{F}}$ be a coherent sheaf of $ {\mathcal{O}}_{X}$-modules and let $ {\mathcal{L}}$ be an ample invertible sheaf over $ X$. Let $ Z_{0} \subseteq X_{0}$ be a closed set. We show that the depth of the hig...

This paper studies the problem of obtaining minimal realizations of linear input/output maps defined over rings. In particular, it is shown that, contrary to the case of systems over fields, it is in general impossible to obtain realizations whose dimension equals the rank of the Hankel matrix. A characterization is given of those (Noetherian) rings over which realizations of such dimensions ca...

Mennicke–Newman lemma for unimodular rows was used by W. van der Kallen to give a group structure on the orbit set $$\frac{Um_{n}(R)}{E_{n}(R)}$$ commutative noetherian ring of dimension $$d\le 2n-4.$$ In this paper, we generalise $$m\times n $$ right invertible matrices

We study finite dimensional representations over some Noetherian algebras a field of characteristic zero. More precisely, we give necessary and sufficient conditions for the category locally to be closed under taking injective hulls extend results known group rings enveloping Ore extensions, Hopf crossed products, affine low Gelfand-Kirillov dimension.

A ring is called a Gelfand ring (pm ring ) if each prime ideal is contained in a unique maximal ideal. For a Gelfand ring R with Jacobson radical zero, we show that the following are equivalent: (1) R is Artinian; (2) R is Noetherian; (3) R has a finite Goldie dimension; (4) Every maximal ideal is generated by an idempotent; (5) Max (R) is finite. We also give the following resu1ts:an ideal...

Abstract This is a general study of twisted Calabi–Yau algebras that are $\mathbb {N}$ -graded and locally finite-dimensional, with the following major results. We prove finite graded algebra if only it separable modulo its radical satisfies one several suitable generalizations Artin–Schelter regularity property, adapted from work Martinez-Villa as well Minamoto Mori. characterize dimension 0 k...

Let (R,m) be a Noetherian local ring, M be a finitely generated R-module of dimension n and a be an ideal of R. In this paper, generalizing the main results of Dibaei and Jafari [3] and Rezaei [8], we will show that if T is a subset of AsshR M, then there exists an ideal a of R such that AttR Hna (M)=T. As an application, we give some relationships between top local cohomology modules and top f...