Embedding Semigroups in Rings - MathOverflow 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 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 Answer by Mark Sapir for Embedding Semigroups in Rings 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>