In search for Noetherian algebras (Q2759667)

The paper presents two new theorems of the author on the problem of Noetherian semigroup algebras. Let \(T\) be a completely \(0\)-simple (completely simple) semigroup. A subsemigroup \(H\) of \(T\) is said to be uniform if and only if \(H\) meets all \(\mathcal H\)-classes of \(T\). In this case a sandwich matrix \(P\) of \(T\) may be chosen so that all the nonzero entries of \(P\) belong to \(H\). Suppose that \(K\) is a field and \(S\) is a monoid with finite Gelfand-Kirillov dimension. If \(M\subset S\) than \(\langle M\rangle\) denotes subsemigroup of \(S\) generated by \(M\). Assume that \(a\langle a,b\rangle\cap b\langle a,b\rangle\neq\emptyset\neq\langle a,b\rangle a\cap\langle a,b\rangle b\) for all \(a,b\in S\).NEWLINENEWLINENEWLINETheorem 3.2 states that in the case \(K[S]\) is left and right Noetherian \(S\) has a finite ideal chain with factors either uniform or nilpotent; additionally if \(Z\) is one of the uniform factors, the sandwich matrix of \(Z\) may be chosen diagonal. Theorem 3.3 shows that, in a way, the converse of Theorem 3.2 also holds. Namely, let \(S\) be a monoid with finite Gelfand-Kirillov dimension and with minimal condition on right ideals. Assume that \(S\) has a finite ideal chain with factors either uniform or nilpotent and suppose that the sandwich matrix of any uniform factor may be chosen diagonal. Then \(K[S]\) is right Noetherian.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00049].