most recent 30 from http://mathoverflow.net 2013-05-23T19:57:58Z http://mathoverflow.net/feeds/question/103450 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103450/embedding-semigroups-in-rings zacarias 2012-07-29T14:09:57Z 2012-07-30T15:59:26Z

<p>Let $S$ be a finite commutative semigroup with identity. Under what conditions (on the semigroup $S$) it is possible to find a ring $R$ such that the multiplicative structure of $R - \{0\}$ is isomorphic to $S$? </p>

http://mathoverflow.net/questions/103450/embedding-semigroups-in-rings/103520#103520 Mark Sapir 2012-07-30T15:59:26Z 2012-07-30T15:59:26Z

<p>$S$ must be a cyclic group of order $p^n-1$ for some prime $p$ and natural $n$. Indeed, since $R\setminus \{0\}$ is a semigroup under multiplication, $R$ does not have zero divisors. Hence $R$ is a division ring. Since $S$ is finite, $R$ is a finite division ring, hence, by Wedderburn, a finite field. Therefore $S$ must be the multiplicative group of a finite field, hence a cyclic group of order $p^n-1$. Note that you do not need to assume that $S$ is commutative. </p>