Our goal is to establish an efficient decomposition of an ideal A of a commutative ring R as an intersection of primal ideals. We prove the existence of a canonical primal decomposition: A = ⋂ P∈XA A(P ), where the A(P ) are isolated components of A that are primal ideals having distinct and incomparable adjoint primes P . For this purpose we define the set Ass(A) of associated primes of the id...

A new class of rings, the class of weakly left localizable rings, is introduced. A ring R is called weakly left localizable if each non-nilpotent element of R is invertible in some left localization SR of the ring R. Explicit criteria are given for a ring to be a weakly left localizable ring provided the ring has only finitely many maximal left denominator sets (eg, this is the case if a ring h...

When R is a local ring with a nilpotent maximal ideal, the Ore extension R[x;σ, δ] will or will not be 2-primal depending on the δ-stability of the maximal ideal of R. In the case where R[x;σ, δ] is 2-primal, it will satisfy an even stronger condition; in the case where R[x;σ, δ] is not 2-primal, it will fail to satisfy an even weaker condition. 1. Background and motivation In [13, Proposition ...

Let $R$ be a commutative ring with identity‎. ‎A proper ideal $P$ of $R$ is a $(n-1,n)$-$Phi_m$-prime ($(n-1,n)$-weakly prime) ideal if $a_1,ldots,a_nin R$‎, ‎$a_1cdots a_nin Pbackslash P^m$ ($a_1cdots a_nin Pbackslash {0}$) implies $a_1cdots a_{i-1}a_{i+1}cdots a_nin P$‎, ‎for some $iin{1,ldots,n}$; ($m,ngeq 2$)‎. ‎In this paper several results concerning $(n-1,n)$-$Phi_m$-prime and $(n-1,n)$-...

An element x of the ring R is called periodic if there exist distinct positive integers m, n such that xm = xn; and x is potent if there exists n > 1 for which xn = x. We denote the set of potent elements by P or P(R), the set of nilpotent elements by N or N(R), the center by Z or Z(R), and the Jacobson radical by J or J(R). The ring R is called periodic if each of its elements is periodic, and...

We introduce the notion of weakly quasi invo-clean rings where every element $ r can be written as r=v+e or r=v-e $, $v\in Qinv(R)$ and e\in Id(R) $. study various properties elements rings. prove that ring R=\prod_{i\in I} R_i all are invo-clean, is if only factors but one invo-clean.