Seminormal ring

In algebra, a seminormal ring is a commutative reduced ring in which, whenever x, y satisfy x 3 = y 2 , there is s with s 2 = x and s 3 = y . This definition was given by Swan (1980) as a simplification of the original definition of Traverso (1970).

A basic example is an integrally closed domain, i.e., a normal ring. For an example which is not normal, one can consider the non-integral ring Z [ x , y ] / x y , or the ring of a nodel curve.

In general, a reduced scheme X can be said to be seminormal if every morphism Y → X which induces a homeomorphism of topological spaces, and an isomorphism on all residue fields, is an isomorphism of schemes.

A semigroup is said to be seminormal if its semigroup algebra is seminormal.