Nikmehr M. J.
Ukr. Mat. Zh. - 2014. - 66, № 11. - pp. 1528–1539
For a monoid M, we introduce strongly M-semicommutative rings obtained as a generalization of strongly semicommutative rings and investigate their properties. We show that if G is a finitely generated Abelian group, then G is torsion free if and only if there exists a ring R with |R| ≥ 2 such that R is strongly G-semicommutative.
Ukr. Mat. Zh. - 2014. - 66, № 9. - pp. 1213–1222
We generalize the concepts of semicommutative, skew Armendariz, Abelian, reduced, and symmetric left ideals and study the relationships between these concepts.
Ukr. Mat. Zh. - 2012. - 64, № 3. - pp. 404-414
We introduce the concept of weak $\alpha$-skew Armendariz ideals and investigate their properties. Moreover, we prove that $I$ is a weak $\alpha$-skew Armendariz ideal if and only if $I[x]$ is a weak $\alpha$-skew Armendariz ideal. As a consequence, we show that $R$ is a weak $\alpha$-skew Armendariz ring if and only if $R[x]$ is a weak $\alpha$-skew Armendariz ring.
Ukr. Mat. Zh. - 2011. - 63, № 4. - pp. 502-512
Let $R$ be a commutative ring with identity, $M$ an $R$-module and $K_1,..., K_n$ submodules of $M$. In this article, we construct an algebraic object, called product of $K_1,..., K_n$. We equipped this structure with appropriate operations to get an $R(M)$-module. It is shown that $R(M)$-module $M^n = M... M$ and $R$-module $M$ inherit some of the most important properties of each other. For example, we show that $M$ is a projective (flat) $R$-module if and only if $M^n$ is a projective (flat) $R(M)$-module.