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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Zhelin Piao
Sung Ju Ryu
Hyo Jin Sung
Sang Jo Yun


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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