Seminormal ring

In algebra, a seminormal ring is a commutative reduced ring in which, whenever x, y satisfy ${\displaystyle x^{3}=y^{2}}$, there is s with ${\displaystyle s^{2}=x}$ and ${\displaystyle s^{3}=y}$. 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 ${\displaystyle \mathbb {Z} [x,y]/xy}$, or the ring of a nodal curve.

In general, a reduced scheme ${\displaystyle X}$ can be said to be seminormal if every morphism ${\displaystyle Y\to X}$ 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.