On weakly local rings
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This article concerns a property of local rings and domains. A ring $R$ is called {\it weakly local} if for every $a\in R$, $a$ is regular or $1-a$ is regular, where a regular element means a non-zero-divisor. We study the structure of weakly local rings in relation to several kinds of factor rings and ring extensions that play roles in ring theory. We prove that the characteristic of a weakly local ring is either zero or a power of a prime number. It is also shown that the weakly local property can go up to polynomial (power series) rings and a kind of Abelian matrix rings.
Abstract
This article concerns a property of local rings and domains. A ring $R$ is called {\it weakly local} if for every $a\in R$, $a$ is regular or $1-a$ is regular, where a regular element means a non-zero-divisor. We study the structure of weakly local rings in relation to several kinds of factor rings and ring extensions that play roles in ring theory. We prove that the characteristic of a weakly local ring is either zero or a power of a prime number. It is also shown that the weakly local property can go up to polynomial (power series) rings and a kind of Abelian matrix rings.
참고문헌 (9)
- E. H. Feller, "Properties of primary noncommutative rings", Trans. Amer. Math. Soc, 89, pp.79~91 (1958)
- K. R. Goodearl, "An Introduction to Noncommutative Noe-therian Rings" (1989), Cambridge University Press
- C.Y. Hong, "Duo property on the monoid of regular elements", ~
- C. Huh, "p.p. rings and generalized p.p.. rings", J. Pure Appl. Algebra, 167, pp.37~52 (2002)
- C. Huh, "Armendariz rings and semicommutative rings", Comm. Algebra, 30, pp.751~761 (2002)
- S. U. Hwang, "On rings whose right annihilators are bounded", Glasgow Math. J, 51, pp.539~559 (2009)
- N. K. Kim, "Armendariz rings and reduced rings", J. Algebra, 223, pp.477~488 (2000)
- J. Lambek, "Lectures on Rings and Modules" (1966), Blaisdell Publishing Company
- J.C. McConnell, "Noncommutative Noetherian Rings" (1987), John Wi-ley & Sons Ltd