Chain conditions on semirings
Let \(S=(S,+, \cdot)\) be an additively commutative semiring with an absorbing zero and \(M=(_SM,+)\) a (left) semimodule over \(S\). An \(S\)-subsemimodule \(U\) of \(M\) is called a \(k\)-subsemimodule \([h\)-subsemimodule] if \(u+x=v\) [\(u+x+z=v+z]\) for \(u,v \in U\) and \(x,z \in M\) imply \(x \in U\), and \(M\) is called artinian [\(k\)-artinian, \(h\)-artinian] if \(M\) satisfies the descending chain condition on \(S\)-subsemimodules [\(k\)-subsemimodules, \(h\)-subsemimodules]. Note that the left ideals of \((S,+, \cdot)\) are the \(S\)-subsemimodules of \((_SS,+)\) and let \(S^n=(_SS^n,+)\) be the direct product of \(n\) copies of \((_SS,+)\). Since the semiring \(M_n(S)\) of all \(n \times n\)-matrices over an artinian semiring \(S\) (with identity) need not be artinian, the following result is of interest: \(M_n(S)\) is artinian [\(k\)-artinian, \(h\)-artinian] iff \(S^n\) is. Further: If \(S\) has a ring of differences \(D(S)\), then \(M_n (S)\) is \(k\)-artinian iff \(D(S)\) is artinian. Other results include the ring \(D(S/\kappa)\) in these considerations, where \(\kappa\) is the least additively cancellative congruence on \((S,+, \cdot)\). In a similar way noetherian [\(k\)-noetherian, \(h\)-noetherian] semirings are defined. One result is: If \(S\) has a ring of differences \(D(S)\), then the polynomial semiring \(S[x]\) is \(k\)-noetherian if \(D(S)\) is noetherian. This generalizes a result of \textit{H. E. Stone} [Trans. Am. Math. Soc. 233, 339-353 (1977; Zbl 0371.16016)], where \(D(S)\) was assumed to have an ideality.
descending chain condition on \(S\)-subsemimodules
scientific article; zbMATH DE number 874316