Ukr. Mat. Zh. - 2014. - 66, № 6. - pp. 767–786
Let $R$ be a ring, let $I$ be an ideal of $R$, and let $n$ be a fixed positive integer. We define and study $I-n$-injective modules and $I-n$-flat modules. Moreover, we define and study left $I-n$-coherent rings, left $I-n$-semihereditary rings, and $I$-regular rings. By using the concepts of $I-n$-injectivity and $I-n$-flatness of modules, we also present some characterizations of the left $I-n$-coherent rings, left $I-n$-semihereditary rings, and $I$-regular rings.
Ukr. Mat. Zh. - 2013. - 65, № 11. - pp. 1476–1481
A ring R is called right almost MGP-injective (or AMGP-injective) if, for any 0 ≠ a ∈ R, there exists an element b ∈ R such that ab = ba ≠ 0 and any right R-monomorphism from abR to R can be extended to an endomorphism of R. In the paper, several properties of these rings are establshed and some interesting results are obtained. By using the concept of right AMGP-injective rings, we present some new characterizations of QF-rings, semisimple Artinian rings, and simple Artinian rings.
Ukr. Mat. Zh. - 2011. - 63, № 10. - pp. 1426-1433
We study MP-injective rings and MGP-injective rings satisfying some additional conditions. Using the concepts of MP-injectivity and MGP-injectivity of rings and modules, we present some new characterizations of QF-rings, semisimple Artinian rings, strongly regular rings, and simple Artinian rings.