let $r=oplus_{nin bbb n_0}r_n$ be a noetherian homogeneous ring with local base ring $(r_0,frak{m}_0)$, $m$ and $n$ two finitely generated graded $r$-modules. let $t$ be the least integer such that $h^t_{r_+}(m,n)$ is not minimax. we prove that $h^j_{frak{m}_0r}(h^t_{r_+}(m,n))$ is artinian for $j=0,1$. also, we show that if ${rm cd}(r_{+},m,n)=2$ and $tin bbb n_0$, then $h^t_{frak{m}_0r}(h^2_{...

A generalization of injective modules (noted GI-modules), distinct from p-injective modules, is introduced. Rings whose p-injective modules are GI are characterized. If M is a left GI-module, E = End(AM), then E/J(E) is von Neumann regular, where J(E) is the Jacobson radical of the ring E. A is semisimple Artinian if, and only if, every left A-module is GI. If A is a left p. p., left GI-ring su...

If R is a commutative ring, then we prove that every finitely generated R-module has a pure-composition series with indecomposable cyclic factors and any two such series are isomorphic if and only if R is a Bézout ring and a CF-ring. When R is a such ring, the length of a pure-composition series of a finitely generated R-module M is compared with its Goldie dimension and we prove that these num...

The class of all Artinian local rings of length at most l is ∀2-elementary, axiomatised by a finite set of axioms Artl. We show that its existentially closed models are Gorenstein, of length exactly l and their residue fields are algebraically closed, and, conversely, every existentially closed model is of this form. The theory Gorl of all Artinian local Gorenstein rings of length l with algebr...

Let R be a (mixed characteristic) Artinian local ring of length l and let X be an n-tuple of variables. This paper provides bounds over the ring R[X] on the degrees of the output of several algebraic constructions in terms of l, n and the degrees of the input. For instance, if I is an ideal in R[X] generated by polynomials gi of degree at most d and if f is a polynomial of degree at most d belo...

Let R be a commutative Noetherian ring, a an ideal of R, and let M be a finitely generated R-module. For a non-negative integer t, we prove that H a(M) is a-cofinite whenever H t a(M) is Artinian and H i a(M) is a-cofinite for all i < t. This result, in particular, characterizes the a-cofiniteness property of local cohomology modules of certain regular local rings. Also, we show that for a loca...

In this paper, we present a general definition of the notion Noetherian and Artinian BL-algebra more comprehensive insight at chain conditions in BL-algebras. We derive some theorems which generalize existence results. give an axiomatic approach to being Artinian, is also applicable other algebraic structures. use theoretical define arithmetic that possible for devices. study, only focus