Study of Noetherian \(\Gamma\)-semirings via operator semirings
The authors study various ascending chain conditions on \(\Gamma\)-semirings by considering the equivalent concepts for the operator semirings. All \(\Gamma\)-semirings considered in this paper possess left and right unities. Central to their investigation is Theorem 2.9 which states that a \(\Gamma\)-semiring \(S\) is right (\(k\)-, \(h\)-) Noetherian if and only if its left operator semiring \(L\) is right \((k\)-, \(h\)-) Noetherian. Cohen's theorem is extended to \(\Gamma\)-semirings, i.e. it is shown that \(S\) is Noetherian if and only if every prime ideal of \(S\) is finitely generated. Various decomposition theorems are then proved for ideals. In particular, it is shown that if \(S\) is right Noetherian, then every proper right ideal of \(S\) can be represented as the intersection of finitely many irreducible right ideals of \(S\).
scientific article; zbMATH DE number 1836954