Algebra and Discrete Mathematics, 2018, Vol. 26, № 1http://dspace.nbuv.gov.ua:80/handle/123456789/1504052026-04-22T10:05:44Z2026-04-22T10:05:44ZType conditions of stable range for identification of qualitative generalized classes of ringsZabavsky, B.V.http://dspace.nbuv.gov.ua:80/handle/123456789/1883812023-02-27T23:27:07Z2018-01-01T00:00:00ZType conditions of stable range for identification of qualitative generalized classes of rings
Zabavsky, B.V.
This article deals mostly with the following question: when the classical ring of quotients of a commutative ring is a ring of stable range 1? We introduce the concepts of a ring of (von Neumann) regular range 1, a ring of semihereditary range 1, a ring of regular range 1, a semihereditary local ring, a regular local ring. We find relationships between the introduced classes of rings and known ones, in particular, it is established that a commutative indecomposable almost clean ring is a regular local ring. Any commutative ring of idempotent regular range 1 is an almost clean ring. It is shown that any commutative indecomposable almost clean Bezout ring is an Hermite ring, any commutative semihereditary ring is a ring of idempotent regular range 1. The classical ring of quotients of a commutative Bezout ring QCl(R) is a (von Neumann) regular local ring if and only if R is a commutative semihereditary local ring.
2018-01-01T00:00:00ZPlanarity of a spanning subgraph of the intersection graph of ideals of a commutative ring I, nonquasilocal caseVadhel, P.Visweswaran, S.http://dspace.nbuv.gov.ua:80/handle/123456789/1883802023-02-26T23:27:54Z2018-01-01T00:00:00ZPlanarity of a spanning subgraph of the intersection graph of ideals of a commutative ring I, nonquasilocal case
Vadhel, P.; Visweswaran, S.
The rings considered in this article are nonzero commutative with identity which are not fields. Let R be a ring. We denote the collection of all proper ideals of R by I(R) and the collection I(R)\{(0)} by I(R)*. Recall that the intersection graph of ideals of R, denoted by G(R), is an undirected graph whose vertex set is I(R)* and distinct vertices I, J are adjacent if and only if I ∩ J ≠ (0). In this article, we consider a subgraph of G(R), denoted by H(R), whose vertex set is I(R)* and distinct vertices I, J are adjacent in H(R) if and only if IJ ≠ (0). The purpose of this article is to characterize rings R with at least two maximal ideals such that H(R) is planar.
2018-01-01T00:00:00ZAlgebraic Morse theory and homological perturbation theorySköldberg, E.http://dspace.nbuv.gov.ua:80/handle/123456789/1883792023-02-26T23:27:48Z2018-01-01T00:00:00ZAlgebraic Morse theory and homological perturbation theory
Sköldberg, E.
We show that the main result of algebraic Morse theory can be obtained as a consequence of the perturbation lemma of Brown and Gugenheim.
2018-01-01T00:00:00ZOn the saturations of submodulesPaudel, L.Tchamna, S.http://dspace.nbuv.gov.ua:80/handle/123456789/1883782023-02-26T23:27:51Z2018-01-01T00:00:00ZOn the saturations of submodules
Paudel, L.; Tchamna, S.
Let R ⊆ S be a ring extension, and let A be an R-submodule of S. The saturation of A (in S) by τ is set A[τ] = {x ∈ S : tx ∈ A for some t ∈ τ}, where τ is a multiplicative subset of R. We study properties of saturations of R-submodules of S. We use this notion of saturation to characterize star operations ⋆ on ring extensions R ⊆ S satisfying the relation (A ∩ B)⋆ = A⋆ ∩ B⋆ whenever A and B are two R-submodules of S such that AS = BS = S.
2018-01-01T00:00:00Z