Let $R\left[x\right]$ be the polynomial ring over a ring $R$ with unity. A polynomial $f\left(x\right)\in R\left[x\right]$ is referred to as a left annihilating content polynomial (left ACP) if there exist an element $r\in R$ and a polynomial $g\left(x\right)\in R\left[x\right]$ such that $f\left(x\right)=rg\left(x\right)$ and $g\left(x\right)$ is not a right zero-divisor polynomial in $R\left[x\right]$. A ring $R$ is referred to as left EM if each polynomial $f\left(x\right)\in R\left[x\right]$ is a left ACP. We observe the structure of left EM rings with various properties, and study the relationships between the one-sided EM condition and other standard ring theoretic conditions. Moreover,...

The aim of this paper is to establish the close connection between prime ideals and torsion theories in a non necessarily commutative noetherian ring. We introduce a new definition of support of a module and characterize some kinds of torsion theories in terms of prime ideals. Using the machinery introduced before, we prove a version of the Mayer-Vietoris Theorem for local cohomology and establish a relationship between the classical dimension and the vanishing of the groups of local cohomology...