Weak primary decomposition theorem for right Noetherian semirings (Q1333749)

The paper deals with semirings \((S,+,\cdot)\) which are additively commutative, have an absorbing zero and an identity. A right ideal \(I\) of \(S\) is called a right \(k\)-ideal if \(a + b \in I\), \(a \in I\) and \(b \in S\) imply \(b\in I\) and weak primary if \(ab \in I\) and \(aI \subseteq I\) imply \(a^ n \in I\) or \(b \in I\). There are even (clearly noncommutative) rings \(S\) containing weak primary right ideals which are not primary. Now let \(S\) be a right Noetherian semiring. The main results of the paper state: Each irreducible right \(k\)-ideal of \(S\) is weak primary. If all right ideals of \(S\) are right \(k\)-ideals, then each right ideal of \(S\) has a weak primary decomposition.