A new class of ideals in semirings (Q1808755)

Let \((S,+,\cdot)\) be an additively commutative semiring with absorbing zero \(0\). Denote by \(P^+(S)\) the set of all \(x\in S\) such that \(nx=nx+x\) for some natural number \(n\). Clearly, \(E^+(S)\subseteq P^+(S)\) for the set \(E^+(S)\) of all idempotent elements of \((S,+)\), and \(P^+(S)=\{0\}\) if \((S,+)\) is cancellative. Moreover, \(E^+(S)=P^+(S)\) if \((S,+)\) is regular. An ideal \(I\) of \(S\) is called a \(p\)-ideal, if \(nx+a=nx+x\) for some \(a\in I\), \(x\in S\) and natural number \(n\) implies \(x\in I\). (Note that for an additively regular semiring this is equivalent with \(P^+(S)\subseteq I\).) The semiring \(S\) is called (left, right) \(p\)-simple if there is no (left, right) \(p\)-ideal \(I\) satisfying \(P^+(S)\subset I\subset S\). Now assume that \(S\) has an identity \(e\not=0\). Then \(S\) is called a \(p\)-semifield, if for each \(a\in S\setminus P^+(S)\) there are \(r,s\in S\) and natural numbers \(n,m\) such that \(ra+ne=ne+e\) and \(as+me=me+e\). Then it is shown that \(S\) is a \(p\)-semifield iff \(S\) is left and right \(p\)-simple. Also the interrelation between maximal \(p\)-ideals \(I\) and \(p\)-semifields \(S/I\) is investigated. Finally, call \(S\) \(p\)-regular, if for each \(a\in S\) there exists some \(b\in S\) and some natural number \(n\) such that \(na+aba=na+a\). (Note that every multiplicatively regular semiring is \(p\)-regular but not conversely.) Now assume that \((S,\cdot)\) is commutative and \((S,+)\) an inverse semigroup. Then some conditions are given such that the matrix semiring \(M_2(S)\) is \(p\)-simple (\(p\)-regular).