\(p\)-ideals in \(p\)-regular semirings
Let \((S,+,\cdot)\) be an additively commutative semiring. If \((S,+)\) is an inverse semigroup, \(S\) is called inversive. The semiring \(S\) is \(p\)-regular if for each \(a\in S\) there is some \(b\in S\) such that \(na+aba=(n+1)a\) for some natural number \(n\). If \(S\) is inversive, this condition reduces to \(a+aba = 2a\). An ideal \(I\) of \(S\) is called a \(p\)-ideal if \(nx+a=(n+1)x\) for \(x\in S\) and \(a\in I\) implies \(x\in I\). An element \(e\in S\) is called \(p\)-idempotent, if \(ne+e^2=(n+1)e\) for some natural number \(n\). In an inversive semiring, this reduces to \(e+e^2=2e\). Some basic statements about ideals and \(p\)-ideals in \(p\)-regular semirings are shown. The main result is: An inversive semiring with identity is \(p\)-regular if every principal left \(p\)-ideal is generated by a \(p\)-idempotent element.
scientific article; zbMATH DE number 1947330