# Strongly Additively Regular Rings and Graphs

• Thomas G. Lucas
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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