The rings considered in this article are commutative rings with identity $1neq 0$. The aim of this article is to define and study the exact annihilating-ideal graph of commutative rings. We discuss the interplay between the ring-theoretic properties of a ring and graph-theoretic properties of exact annihilating-ideal graph of the ring.

a polynomial   1 2 ( , , , ) n f x x x is called multilinear if it is homogeneous and linear in every one of its variables. in the present paper our objective is to prove the following result: let   r be a prime k-algebra over a commutative ring   k with unity and let 1 2 ( , , , ) n f x x x be a multilinear polynomial over k. suppose that   d is a nonzero derivation on r such that ...

the aim of this paper is to study orders over a valuation ring v with arbitrary rank in acentral simple f-algebra q. the relation between all of the orders is explained with a diagram. it is thenshown that inside bezout order, extremal v-orders are precisely semi-hereditary. in the last section, theeffect of henselization on maximal and semi-hereditary orders is examined.

let $r$ be a commutative ring with identity and $mathbb{a}(r)$ be the set   of ideals of $r$ with non-zero annihilators. in this paper, we first introduce and investigate the principal ideal subgraph of the annihilating-ideal graph of $r$, denoted by $mathbb{ag}_p(r)$. it is a (undirected) graph with vertices $mathbb{a}_p(r)=mathbb{a}(r)cap mathbb{p}(r)setminus {(0)}$, where   $mathbb{p}(r)$ is...

this is a survey of some recent applications of abstract algebra, and in particular group rings, to the `communications' areas.

In this paper, we introduce the new notion of n-f-clean rings as a generalization of f-clean rings. Next, we investigate some properties of such rings. We prove that $M_n(R)$ is n-f-clean for any n-f-clean ring R. We also, get a condition under which the denitions of n-cleanness and n-f-cleanness are equivalent.

atropine has been used to block cholinergic neurotransmission in basic research. large doses of atropine cause vasodilation of the blood vessels in the skin. this effect is apparently unconnected with the antimuscarinic activity of atropine and seems to be due to a direct action on the blood vessels. it has been suggested that atropine blocks muscarinic receptors at low doses and it induces the...

this study is attempted to compare the number of rings and circular scales that are formed during various ages on liza saliens. sampling was carried out monthly during 2012-2013 using beach sine fishing nets. 102 specimens of different sizes were collected and transferred to the laboratory in boxes containing ice. the age of fish was determined by the number of annual rings as well as, the numb...

This study is attempted to compare the number of rings and circular scales that are formed during various ages on Liza saliens. Sampling was carried out monthly during 2012-2013 using beach sine fishing nets. 102 specimens of different sizes were collected and transferred to the laboratory in boxes containing ice. The age of fish was determined by the number of annual rings as well as, the numb...