Let be a ring. We use mod (resp. mod ) to denote the category of finitely generated left -modules (resp. right -modules). We always assume that and are Artinian algebras and is a faithfully balanced self-orthogonal bimodule, that is, satisfies the following conditions: (1) is in mod and is in mod ; (2) the natural maps → End op and → End are isomorphisms; (3) Ext = 0 and Ext = 0 for any i ≥ 1. ...

let $r$ be an arbitrary ring with identity and $m$ a right $r$-module with $s=$ end$_r(m)$. the module $m$ is called {it rickart} if for any $fin s$, $r_m(f)=se$ for some $e^2=ein s$. we prove that some results of principally projective rings and baer modules can be extended to rickart modules for this general settings.

We define generalized Koszul modules and rings develop a theory for N-graded with the degree zero part noetherian semiperfect. This specializes to classical graded artinian semisimple developed by Beilinson-Ginzburg-Soergel ungraded semiperfect Green Martinéz-Villa. Let A be left finite ring generated in 1 A0 semiperfect, J its Jacobson radical. By dual of we mean Yoneda Ext Ext_A•(A/J,A/J). If...

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...

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

In this paper we will study artinian quotients A = R/I of the polynomial ring R = k[x1, ..., xr], where k is a field of characteristic zero, the xi’s all have degree 1 and I is a homogeneous ideal of R. These rings are often referred to as standard graded artinian algebras. Before explaining the main results of this work, we establish some of the notation we will use: the h-vector of A is h(A) ...

In this paper, we describe the maximal bounded Z-filtrations of Artinian semisimple rings. These turn out to be the filtrations associated to finite Z-gradings. We also consider simple Artinian rings with involution, in characteristic 6= 2, and we determine those bounded Z-filtrations that are maximal subject to being stable under the action of the involution. Finally, we briefly discuss the an...

Let (R,m) be a commutative Noetherian complete local ring, a an ideal of R, and A an Artinian R-module with N-dim A = d. We prove that if d > 0, then Cosupp(H d−1(A)) is finite and if d ≤ 3, then the set Coass(H i (A)) is finite for all i. Moreover, if either d ≤ 2 or the cohomological dimension cd(a) = 1 then H i (A) is a-coartinian for all i; that is, Torj (R/a,H a i (A)) is Artinian for all ...

In this paper, we study the problem when a finitely generated torsionless module is projective. Let Λ be an Artinian local algebra with radical square zero. Then a finitely generated torsionless Λ-module M is projective if Ext Λ (M,M) = 0. For a commutative Artinian ring Λ, a finitely generated torsionless Λ-module M is projective if the following conditions are satisfied: (1) Ext Λ (M,Λ) = 0 f...