The basic properties of multiplication algebras of nonassociative algebras over rings are introduced, including a discussion of multiplication algebras of tensor products of algebras. A characterization of semisimple artinian multiplication algebras is given along with a discussion of the nature of the simple factors of a multiplication algebra modulo its Jacobson radical. A criterion distingui...

In this paper we are concerned with the finiteness property of Extindices of several ring extensions. In this direction, we introduce some conjectures and discuss the relationship of them. Also we give affirmative answers to these conjectures in some special cases. Furthermore, we prove that the trivial extension of an Artinian local ring by its residue class field is always of finite Ext-index...

A local ring is an associative with unique maximal ideal. We associate each Artinian singleton basis four invariants (positive integers) p,n,s,t. The purpose of this article to describe the structure such rings and classify them (up isomorphism) same invariants. Every can be constructed over its coefficient subring by a certain polynomial called associated polynomial. These polynomials play sig...

Let R be a commutative ring with non-zero identity. The annihilator-inclusion ideal graph of R , denoted by ξR, is a graph whose vertex set is the of allnon-zero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacentif and only if either Ann(I) ⊆ J or Ann(J) ⊆ I. In this paper, we investigate the basicproperties of the graph ξR. In particular, we showthat ξR is a connected grap...

Let A⊆B be a ring extension and G set of A-submodules B. We introduce class closure operations on (which we call multiplicative (A,B,G)) that generalizes the classes star, semistar semiprime operations. study how Mult(A,B,G) these varies when A, B or vary, behaves under homomorphisms. As an application, show to reduce star analytically unramified one-dimensional Noetherian domains closures fini...

‎Let $I$ be an ideal in a regular local ring $(R,n)$‎, ‎we will find‎ ‎bounds on the first and the last Betti numbers of‎ ‎$(A,m)=(R/I,n/I)$‎. ‎if $A$ is an Artinian ring of the embedding‎ ‎codimension $h$‎, ‎$I$ has the initial degree $t$ and $mu(m^t)=1$‎, ‎we call $A$ a {it $t-$extended stretched local ring}‎. ‎This class of‎ ‎local rings is a natural generalization of the class of stretched ...