Suppose that G is a finite group and k field of characteristic $$p>0$$ . We consider the complete cohomology ring $${\mathcal {E}}_M^* = \sum _{n \in {\mathbb Z}} \widehat{{\text {Ext}}}^n_{kG}(M,M)$$ show has two distinguished ideals $$I^* \subseteq J^* {\mathcal {E}}_M^*$$ such $$I^*$$ bounded above in degrees, {E}}_M^*/J^*$$ below degree $$J^*/I^*$$ eventually periodic with terms dimension. ...

The class of rings $\mathcal{J}=\{A (A,\circ)$ forms a group$\}$ radical and it is called the Jacobson class. For any ring $A$, $\mathcal{J}(A)$ $A$ defined as largest ideal which belongs to $\mathcal{J}$. In fact, one most important classes since used widely in another branch abstract algebra, for example, construct two-sided brace. On other hand, every Morita context $T=\begin{pmatrix} R & V ...

let r be an associative ring with unity. an element a in r is said to be r-clean if a = e+r, where e is an idempotent and r is a regular (von neumann) element in r. if every element of r is r-clean, then r is called an r-clean ring. in this paper, we prove that the concepts of clean ring and r-clean ring are equivalent for abelian rings. further we prove that if 0 and 1 are the only idempotents...

let r be an associative ring with unity. an element a in r is said to be r-clean if a = e+r, where e is an idempotent and r is a regular (von neumann) element in r. if every element of r is r-clean, then r is called an r-clean ring. in this paper, we prove that the concepts of clean ring and r-clean ring are equivalent for abelian rings. further we prove that if 0 and 1 are the only idempotents...

In this paper, we study conditions guaranteeing that a product of ideals defines Golod ring. We show for 3 3 -dimensional regular local ring (...