Abstract:Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
“…Furthermore, we generalize [9, Theorem 2.1] and prove that every J-stable ring is strongly completeable. This also extend [8,Theorem] to much wider class of rings (maybe with zero divisors).…”
Section: Introductionmentioning
confidence: 52%
“…Also every ring having almost stable range 1 is strongly completable. As a consequence, we have Corollary 4.14 [8,Theorem]. Every Dedekind domain is strongly completable.…”
A commutative ring R is J-stable provided that R/aR has stable range 1 for all a ∈ J(R). A commutative ring R in which every finitely generated ideal is principal is called a Bézout ring. A ring R is an elementary divisor ring provided that every matrix over R admits a diagonal reduction. We prove that a J-stable ring is a Bézout ring if and only if it is an elementary divisor ring. Further, we prove that every J-stable ring is strongly completable. Various types of J-stable rings are provided. Many known results are thereby generalized to much wider class of rings, e.g.
“…Furthermore, we generalize [9, Theorem 2.1] and prove that every J-stable ring is strongly completeable. This also extend [8,Theorem] to much wider class of rings (maybe with zero divisors).…”
Section: Introductionmentioning
confidence: 52%
“…Also every ring having almost stable range 1 is strongly completable. As a consequence, we have Corollary 4.14 [8,Theorem]. Every Dedekind domain is strongly completable.…”
A commutative ring R is J-stable provided that R/aR has stable range 1 for all a ∈ J(R). A commutative ring R in which every finitely generated ideal is principal is called a Bézout ring. A ring R is an elementary divisor ring provided that every matrix over R admits a diagonal reduction. We prove that a J-stable ring is a Bézout ring if and only if it is an elementary divisor ring. Further, we prove that every J-stable ring is strongly completable. Various types of J-stable rings are provided. Many known results are thereby generalized to much wider class of rings, e.g.
“…Many known results are thereby generalized to much wider class of rings, e.g. [4,Theorem 14], [8,Theorem 3.7], [9,Theorem ], [11, Theorem 1.2.13 and Theorem 1.2.21] and [12,Theorem 32].…”
Section: Introductionmentioning
confidence: 99%
“…Thus, every VNL ring is strongly completable. As every Dedekind domain has almost stable range 1, it follows from Theorem 4.1 that every Dedekind domain is strongly completable[9, Theorem ].…”
A commutative ring R is J-stable provided that R/aR has stable range 1 for all a ∈ J(R). A commutative ring R in which every finitely generated ideal is principal is called a Bézout ring. A ring R is an elementary divisor ring provided that every matrix over R admits a diagonal reduction. We prove that a J-stable ring is a Bézout ring if and only if it is an elementary divisor ring. Further, we prove that every J-stable ring is strongly completable. Various types of J-stable rings are provided. Many known results are thereby generalized to much wider class of rings, e.g.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.