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.