Fully idempotent semirings
The paper deals with semirings \((A,+,\cdot)\) which are additively commutative and contain an absorbing zero. Such a semiring is called fully idempotent if \(I^ 2 = I\) holds for each ideal \(I\) of \((A,+,\cdot)\). As for rings, this is equivalent with \(I\cap J = IJ\) for all ideals \(I\) and \(J\), and the author shows that the lattice of all ideals of a fully idempotent semiring is Brouwerian and hence distributive. The main result states that \((A,+,\cdot)\) is fully idempotent iff each ideal \(I\) of \((A,+,\cdot)\) is the intersection of all prime ideals containing \(I\). Similar statements are proved for ideals of a semigroup \((S,\cdot)\).
scientific article; zbMATH DE number 556449