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Strongly Additively Regular Rings and Graphs

  • Thomas G. LucasEmail author
Chapter
Part of the Trends in Mathematics book series (TM)

Abstract

A commutative ring R is said to be additively regular if for each pair of elements \(f,g\in R\) with f regular, there is an element \(t\in R\) such that \(g+ft\) is regular. For any commutative ring R, the polynomial ring \(R[{\scriptstyle \mathrm {X}}]\) is additively regular, moreover if \(deg(g)<n\), then \(g+f{\scriptstyle \mathrm {X}}^n\) is regular when \(f\in R[x]\) is regular. We introduce several stronger types of additively regular rings where the choice for t is restricted: R is strongly additively regular if for each pair of elements \(f,g\in R\) with f regular and g a zero divisor, there is a regular element \(t\in R\) such that \(g+ft\) is regular; R is very strongly additively regular if for each pair of elements \(h,k\in R\) with h regular, there is a regular element \(s\in R\) such that \(k+hs\) is regular. Even stronger are strongly u-additively regular and very strongly u-additively regular, for these the “t” is further restricted to being a unit of R.

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Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of North Carolina CharlotteCharlotteUSA

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