Injective and automorphism-invariant non-singular modules

Every automorphism-invariant non-singular right A-module is injective if and only if the factor ring of the ring A with respect to its right Goldie radical is a right strongly semiprime ring.     

Rings without infinite sets of noncentral orthogonal idempotents

For any ring A, there exist a Bezout ring R and an idempotent e ∈ R with A ≅ eRe. Every module over any ring is a direct summand of an endo-Bezout module. Over any ring, every free module of infinite rank is an endo-Bezout module.     

The Jacobson radical of the Laurent series ring

For a large class of rings A, including all rings with right Krull dimension, it is proved that for every automorphism ϕ of the ring A, the Jacobson radical of the skew Laurent series ring A((x, ϕ)) is nilpotent and coincides with N((x, ϕ)), where N is the prime radical of the ring A. If A/N is a ring of bounded index, then the Jacobson radical of the Laurent series ring A((x)) coincides with N((x)). __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 2, pp. 209–215, 2006.     

Distributive and Semihereditary Rings

Let A be a right and left distributive ring. For a positive integer n, we obtain a criterion of projectivity of all n-generated right ideals of the ring A and a criterion of the right semi-heredity of the ring A.__________Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 1, pp. 253–258, 2003.