A study of a new class of ideals in semiring (Q2724008)

A new class of ideals in additively commutative semirings is introduced. An ideal \(I\) of a semiring \(S\) is called a \(p\)-ideal if for some \(x\in S\), \(n\in\mathbb{N}\), \(nx+a=(n+1)x\) and \(a\in I\) implies \(x\in I\) (\(\mathbb{N}\) is the set of natural numbers). For any subsemiring \(R\) of a semiring \(S\) is defined \(\overline R=\{x\in S\mid a+nx=(n+1)x\) for some \(n\in\mathbb{N}\), \(a\in R\}\).NEWLINENEWLINENEWLINEFor any ideal \(I\) of \(S\) there exists a smallest \(p\)-ideal \(I'\) of \(S\) containing \(I\) (Proposition 2.10). A semiring \(S\) is called \(p\)-regular, if for each \(a\in S\) there exists some \(b\in S\) such that \(na+aba=(n+1)a\) for some \(n\in\mathbb{N}\). This notation generalizes ordinary regularity (Proposition 3.6). Similarly, a \(p\)-idempotent element \(e\in S\) is defined as follows: \(ne+e^2=(n+1)e\) for some \(n\in\mathbb{N}\).NEWLINENEWLINENEWLINEUsing the previous notations several characterisations of \(p\)-regularity are obtained.NEWLINENEWLINENEWLINEIn Lemma 3.19 it should be stated that the semiring \(R\) is inversive.