We introduce the class of “right almost V-rings” which is properly between the classes of right V-rings and right good rings. A ring R is called a right almost V-ring if every simple R-module is almost injective. It is proved that R is a right almost V-ring if and only if for every R-module M, any complement of every simple submodule of M is a direct summand. Moreover, R is a right almost V-rin...

In this paper a generalized version of the Bass formula is proved for finitely generated modules of finite Gorenstein injective dimension over a commutative noetherian ring.

We find necessary and sufficient conditions for a complete local (Noetherian) ring containing the rationals to be completion of countable excellent domain. Furthermore, we noncatenary domain, as well it unique factorization

Let R be a commutative Noetherian ring and let M be a nitely generated R-module. If I is an ideal of R generated by M-regular sequence, then we study the vanishing of the rst Tor functors. Moreover, for Artinian modules and coregular sequences we examine the vanishing of the rst Ext functors.

We show the existence of a first order theory Cmd,e whose Noetherian models are precisely the local Cohen-Macaulay rings of dimension d and multiplicity e. The completion of a model of Cmd,e is again a model and is moreover Noetherian. If R is an equicharacteristic local Gorenstein ring of dimension d and multiplicity e with algebraically closed residue field and if the Artin Approximation Prop...

Let R be a Noetherian local ring and Ω an arbitrary R-module of finite depth and finite projective dimension. The flat dimension of Ω is at least depth(R)−depth(Ω) with equality in the following cases: (i) Ω is finitely generated over some Noetherian local R-algebra S; (ii) dim(R) = 1; (iii) dim(R) = 2 and Ω is separated; (iv) R is CohenMacaulay, dim(R) = 3 and Ω is complete.

An easy consequence of this is that a left Noetherian (respectively left Artinian) ring which is finitely generated over its center is right Noetherian (respectively right Artinian). Theorem 1 follows easily from Theorem 2, which gives a partial converse to the following standard fact: IfR C S are rings, and ifQ is an injective R-module, then Hom~(S, Q) is an injective S-module (this follows, f...

Skolem and Nullstellensatz properties are analogues of the weak Nullstellensatz and Hilbert’s Nullstellensatz, respectively, for the ring of integervalued polynomials in several indeterminates Int(D) = {f ∈ K[x1, . . . , xn] | f(D) ⊆ D}, where D is a domain and K its quotient field. We show their equivalence when D is a Noetherian domain and extend the criterion of Brizolis and Chabert for Int(...