We initiate the study of 1-torsion of finite modules over two-sided noetherian semiperfect rings. In particular, we give a criterion for determining when the 1-torsion submodule contains minimal generators of the module. We also provide an explicit construction for a projective cover of the submodule generated by the torsion elements in the top of the module. Some of the obtained results hold w...

An R-module is called semi-endosimple if it has no proper fully invariant essential submodules. For a quasi-projective retractable module MR we show that M is finitely generated semi-endosimple if and only if the endomorphism ring of M is a finite direct sum of simple rings. For an arbitrary module M , conditions equivalent to the semi-endosimplicity of its quasi-injective hull are found. As co...

We consider algebras over a field K defined by a presentation K〈x1, . . . , xn : R〉, where R consists of ( n 2 ) square-free relations of the form xixj = xkxl with every monomial xixj , i 6= j, appearing in one of the relations. Certain sufficient conditions for the algebra to be noetherian and PI are determined. For this, we prove more generally that right noetherian algebras of finite Gelfand...

In abstract algebra, a structure is said to be Noetherian if it does not admit infinite strictly ascending chains of congruences. In this paper, we adapt this notion to first-order logic by defining the class of Noetherian theories. Examples of theories in this class are Linear Arithmetics without ordering and the empty theory containing only a unary function symbol. Interestingly, it is possib...

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...