Advances in Commutative Algebra pp 113-134 | Cite as

# Strongly Additively Regular Rings and Graphs

## 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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