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On a class of Noetherian algebras. (English) Zbl 0943.20068

Let \(G\) be a group, and let \(M_n(G)\) be the semigroup of all monomial \(n\times n\)-matrices over \(G^0=G\cup\{0\}\) having no more than one non-zero entry in any row and in any column. Let \(K\) be a field, and let \(G\) be a polycyclic-by-finite group. It is known that in this case \(K[G]\) is Noetherian. The authors extend this result to a class of submonoids of \(M_n(G)\), namely they show that, for a submonoid \(S\) of \(M_n(G)\), \(K[S]\) is right Noetherian if and only if \(S\) satisfies the ascending chain condition on right ideals (Theorem 1.1). Another interesting result of the paper pertains to the Malcev nilpotent semigroups. Let \(P\) be an ideal of \(K[S]\), and let \(\sim_P\) denote the congruence on \(S\) induced by the natural epimorphism \(K[S]\to K[S]/P\): if \(x,y\in S\), then \(x\sim_Py\) if and only if \(x-y\in P\). A semigroup \(S\) isomorphic to a subsemigroup of \(M_n(G)\) for certain \(G\) is called a monomial semigroup. Suppose that \(S\) is a finitely generated Malcev nilpotent monoid with ascending chain condition on right ideals, and let \(P\) be a prime ideal of \(K[S]\). It is shown that in this case \(S/\sim_P\) is a monomial semigroup over a nilpotent group; in particular, \(K[S/\sim_P]\) is (right and left) Noetherian, and so \(K[S]/P\) is Noetherian (Theorem 1.2).

MSC:

20M25 Semigroup rings, multiplicative semigroups of rings
16S36 Ordinary and skew polynomial rings and semigroup rings
16P40 Noetherian rings and modules (associative rings and algebras)
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