Bezout Rings,Polynomials, and Distributivity

Let A be a ring, be an injective endomorphism of A, and let be the right skew polynomial ring. If all right annihilator ideals of A are ideals, then R is a right Bezout ring is a right Rickartian right Bezout ring, (e)=e for every central idempotent eA, and the element (a) is invertible in A for every regular aA. If A is strongly regular and n 2, then R/x ⁿ R is a right Bezout ring R/x ⁿ R is a right distributive ring R/x ⁿ R is a right invariant ring (e)=e for every central idempotent eA. The ring R/x ² R is right distributive R/x ⁿ R is right distributive for every positive integer n A is right or left Rickartian and right distributive, (e)=e for every central idempotent eA and the (a) is invertible in A for every regular aA. If A is a ring which is a finitely generated module over its center, then A[x] is a right Bezout ring A[x]/x ² A[x] is a right Bezout ring A is a regular ring.     

Completely integrally closed modules and rings. III

We study rings A over which all cyclic right modules are completely integrally closed. The complete answer is obtained if either A is a semiperfect ring or each ring direct factor of A that is a domain is right bounded.     

Completely integrally closed modules and rings. II

A right or left uniserial domain A is a completely integrally closed right A-module if and only if A is an invariant uniserial domain with at most two prime ideals.     

Laurent rings

This is a study of ring-theoretic properties of a Laurent ring over a ring A, which is defined to be any ring formed from the additive group of Laurent series in a variable x over A, such that left multiplication by elements of A and right multiplication by powers of x obey the usual rules, and such that the lowest degree of the product of two nonzero series is not less than the sum of the lowest degrees of the factors. The main examples are skew-Laurent series rings A((x; ϕ)) and formal pseudo-differential operator rings A((t ⁻¹; δ)), with multiplication twisted by either an automorphism ϕ or a derivation δ of the coefficient ring A (in the latter case, take x = t ⁻¹). Generalized Laurent rings are also studied. The ring of fractional n-adic numbers (the localization of the ring of n-adic integers with respect to the multiplicative set generated by n) is an example of a generalized Laurent ring. Necessary and/or sufficient conditions are derived for Laurent rings to be rings of various standard types. The paper also includes some results on Laurent series rings in several variables. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 3, pp. 151–224, 2006.     

Comultiplication Modules over Noncommutative Rings

Comultiplication modules over not necessarily commutative rings are studied.     

Homomorphisms close to regular and their applications

This paper contains new and known results on homomorphisms that are close to regular. The main results are presented with proofs.