Semiboolean semigroup rings (I). Regular semigroup case

The semibooleanness is a clean property of rings. The aim of this paper is to determine when a semigroup ring is semiboolean. The conditions on a coefficient ring together with conditions on a [Formula: see text]-semigroup (regular semigroup) are completely characterized when the corresponding semigroup ring is semiboolean. Moreover, we give criteria for a regular semigroup ring to be uniquely clean.