## 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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### References:

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