Korean J. Math. Vol. 28 No. 1 (2020) pp.65-73
DOI: https://doi.org/10.11568/kjm.2020.28.1.65

On weakly local rings

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Published: Mar 30, 2020
Keywords:
weakly local ring, local ring, ring of integers modulo n, characteristic, domain, matrix ring, polynomial ring, Abelian ring
Mathematics Subject Classification:
16Lxx, 16S36, 16U10, 16D25

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Zhelin Piao
Department of Mathematics, Yanbian University Yanji 133002, China
Sung Ju Ryu
Department of Mathematics, Pusan National University Busan 46241, Korea
Hyo Jin Sung
Department of Mathematics, Pusan National University Busan 46241, Korea
Sang Jo Yun
Department of Mathematics, Dong-A University Busan, 49315, Korea

Abstract

This article concerns a property of local rings and domains. A ring $R$ is called 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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