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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.
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참고문헌 (9)
참고문헌 신청
E. H. Feller / 1958 / Properties of primary noncommutative rings / Trans. Amer. Math. Soc 89:79~91
K. R. Goodearl / 1989 / An Introduction to Noncommutative Noe-therian Rings / Cambridge University Press
C.Y. Hong / Duo property on the monoid of regular elements
C. Huh / 2002 / p.p. rings and generalized p.p.. rings / J. Pure Appl. Algebra 167:37~52
C. Huh / 2002 / Armendariz rings and semicommutative rings / Comm. Algebra 30:751~761
K. R. Goodearl / 1989 / An Introduction to Noncommutative Noe-therian Rings / Cambridge University Press
C.Y. Hong / Duo property on the monoid of regular elements
C. Huh / 2002 / p.p. rings and generalized p.p.. rings / J. Pure Appl. Algebra 167:37~52
C. Huh / 2002 / Armendariz rings and semicommutative rings / Comm. Algebra 30:751~761
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