Fully prime semirings (Q2718036)

Let \((S,+,\cdot)\) be an additively commutative semiring with absorbing zero 0 and identity \(1\not=0\). An ideal \(P\not=S\) is prime if \(IJ\subseteq P\) implies \(I\subseteq P\) or \(J\subseteq P\) for all ideals \(I,J\) of \(S\). The semiring \(S\) is called fully prime if every ideal \(P\not=S\) is prime. It is shown that \(S\) is fully prime iff every ideal \(I\) of \(S\) is idempotent, i.e. \(II=I\), and the set \(\text{ideal}(S)\) of all ideals of \(S\) is totally ordered by inclusion. The authors also characterize those factor semirings \(\mathbb{N}_0/\kappa\) of the natural numbers \(\mathbb{N}_0\) which are fully prime, and they give a description of all \(k\)-ideals of \(\mathbb{N}_0/\kappa\).