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Structure of Chinese algebras. (English) Zbl 1246.16022

The Chinese monoid ${M}_{n}$ of rank $n$ is generated by ${a}_{1},{a}_{2},\cdots ,{a}_{n}$ subject to the relations

${a}_{j}{a}_{i}{a}_{k}={a}_{j}{a}_{k}{a}_{i}={a}_{k}{a}_{j}{a}_{i},\phantom{\rule{1.em}{0ex}}i\le k\le j·$

The monoid ${M}_{n}$ is infinite and has polynomial growth.

In the paper under review the authors study the structure of the monoid algebra $K\left[{M}_{n}\right]$ of ${M}_{n}$ over a field $K$. The authors show that $K\left[{M}_{n}\right]$ has only finitely many prime ideals and completely describe them using certain homogeneous congruences on ${M}_{n}$. Further, it is shown that the prime radical of $K\left[{M}_{n}\right]$ coincides with the Jacobson radical. As a consequence the authors derive a new representation of ${M}_{n}$ as a submonoid of the product ${B}^{k}×{ℤ}^{l}$ for some $k,l\in ℕ$, where $B$ denotes the bicyclic monoid, and show that ${M}_{n}$ satisfies a nontrivial identity.

##### MSC:
 16S36 Ordinary and skew polynomial rings and semigroup rings 16S15 Finite generation, finite presentability, normal forms 20M25 Semigroup rings, multiplicative semigroups of rings 16N60 Prime and semiprime associative rings 20M05 Free semigroups, generators and relations, word problems 16D25 2-sided ideals (associative rings and algebras)