Topic
Ore condition
About: Ore condition is a research topic. Over the lifetime, 108 publications have been published within this topic receiving 1059 citations.
Papers published on a yearly basis
Papers
1 Feb 1972
TL;DR: Theorem 1.1 as discussed by the authors shows that the group-ring of some generalized free products of groups has no zero divisors, and the right and left Ore conditions are equivalent for group-rings.
Abstract: Let ZG1 and ZG2 be the integral group rings of groups G1 and G2 with a common normal subgroup H and let K be a subgroup of H. Let G be the free product of G1 and G2 amalgamating K. If ZG1 and ZG2 are integral domains and if ZH has the Ore condition then ZG is again an integral domain. In this paper we apply a theorem of P. M. Cohn [1] to show that the group-ring of some generalized free products of groups has no zero divisors. Recall that a ring R without zero divisors has the right Ore condition if any two nonzero elements of R have a common nonzero right multiple. In this situation R has a uniquely determined skew field D of right quotients: every element of D is of the form xy-1 with x, y'cR. (See e.g. [2, Theorem 1.3].) Let us agree, by abus de language, to say that the group G has no zero divisors if the group-ring ZG has no zero divisors. Let H be a subgroup of the group G without zero divisors, and suppose ZG has the right Ore condition. (We note in passing that the right and left Ore condition are equivalent for group-rings.) Then ZH also has the right Ore condition. For ZG is a free right ZH module freely generated by a right transversal of G modulo H. Thus if x, y are in ZH there are nonzero elements t, u in ZG with xt=yu. We need only consider this equation coset per coset to find nonzero elements t' and u' in ZH with xt'=yu'. This said, we may proceed to our results. THEOREM 1. Let Gi (i= 1, 2) be a group without zero divisors, and let Hi be a normal subgroup of Gi such that ZHi has the right Ore condition. Let K be a common subgroup of H1 and H2 and let G be the generalizedfree product of G1 and G2 amalgamating K. Then ZG has no zero divisors. PROOF. Let Di be the fields of quotients of ZHi, and consider the abelian group Ri=ZGi ?ZH, Di. We turn Ri into a ring by defining (g1 (0 11)(g2 (0 121) = g9g2 (0 (g21112) -11-1 Received by the editors January 15, 1971. AMS 1970 subject class/ifcations. Primary 16A26; Secondary 20E30.
71 citations
TL;DR: A computer algebra library for basic operations in an arbitrary Ore ring can be used as a basis for various algorithms in Ore rings, in particular, in differential, shift, and q-shift rings.
Abstract: We present some algorithms related to rings of Ore polynomials (or, briefly, Ore rings) and describe a computer algebra library for basic operations in an arbitrary Ore ring. The library can be used as a basis for various algorithms in Ore rings, in particular, in differential, shift, and q-shift rings.
60 citations
TL;DR: It is proved that G is A-connected for any abelian group A with |A|>=3 if and only if G is not one of the 12 completely described graphs.
Abstract: Let G be a simple graph satisfying the Ore-condition with |V(G)|>=3 In this paper, we prove that G is A-connected for any abelian group A with |A|>=3 if and only if G is not one of the 12 completely described graphs
46 citations
TL;DR: It is proved that with six exceptions, all graphs satisfying the Ore condition have a nowhere-zero 3-flow.
Abstract: Let $G$ be a simple graph on $n$ vertices, $n\geq 3$. It is well known that if $G$ satisfies the Ore condition that $d(x)+d(y)\geq n$ for every pair of nonadjacent vertices $x$ and $y$, then $G$ has a Hamiltonian circuit, which implies that $G$ has a nowhere-zero 4-flow. But it is not necessary for $G$ to have a nowhere-zero 3-flow. In this paper, we prove that with six exceptions, all graphs satisfying the Ore condition have a nowhere-zero 3-flow. More precisely, if $G$ is a graph on $n$ vertices, $n\geq 3$, in which $d(x)+d(y)\geq n$ for every pair of nonadjacent vertices $x$ and $y$, then $G$ has no nowhere-zero 3-flow if and only if $G$ is one of six completely described graphs.
42 citations
TL;DR: In this article, it was shown that if S is an anti-archimedean subset of D, then D is a t-locally S-Noetherian domain if and only if the polynomial ring D(X)Nv is a T-Nagata ring of S-noetherian domains.
Abstract: Let D be an integral domain, t be the so-called t-operation on D; and S be a (not necessarily saturated) multiplicative subset of D. In this paper, we study the Nagata ring of S-Noetherian domains and locally S-Noetherian domains. We also inves- tigate the t-Nagata ring of t-locally S-Noetherian domains. In fact, we show that if S is an anti-archimedean subset of D, then D is an S-Noetherian domain (respectively, locally S-Noetherian domain) if and only if the Nagata ring D(X)N is an S-Noetherian domain (respectively, locally S-Noetherian domain). We also prove that if S is an anti-archimedean subset of D, then D is a t-locally S-Noetherian domain if and only if the polynomial ring D(X) is a t-locally S-Noetherian domain, if and only if the t-Nagata ring D(X)Nv is a t-locally S-Noetherian domain.
39 citations
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Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 4 |
| 2020 | 3 |
| 2017 | 5 |
| 2016 | 8 |
| 2015 | 5 |
| 2014 | 2 |