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Zbl 1246.16022
Jaszuńska, Joanna; Okniński, Jan
Structure of Chinese algebras. (English)
[J] J. Algebra 346, No. 1, 31-81 (2011). ISSN 0021-8693

The Chinese monoid $M_n$ of rank $n$ is generated by $a_1,a_2,\dots,a_n$ subject to the relations $$a_ja_ia_k=a_ja_ka_i=a_ka_ja_i,\quad i\le k\le j.$$ The monoid $M_n$ is infinite and has polynomial growth.\par In the paper under review the authors study the structure of the monoid algebra $K[M_n]$ of $M_n$ over a field $K$. The authors show that $K[M_n]$ 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 $K[M_n]$ coincides with the Jacobson radical. As a consequence the authors derive a new representation of $M_n$ as a submonoid of the product $B^k\times\mathbb Z^l$ for some $k,l\in\mathbb N$, where $B$ denotes the bicyclic monoid, and show that $M_n$ satisfies a nontrivial identity.
[Volodymyr Mazorchuk (Uppsala)]

MSC 2000:: *16S36 Ordinary and skew polynomial rings and semigroup rings
16S15 Finite generation, finite presentability
20M25 Semigroup rings, multiplicative semigroups of rings
16N60 Prime and semiprime assoc. rings
20M05 Free semigroups
16D25 2-sided ideals (assoc. rings and algebras)

Keywords: Chinese monoids; monoid algebras; prime ideals; homogeneous congruences; radicals; representations; presentations

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