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.

References

• Swan, Richard G. (1980), "On seminormality", Journal of Algebra, 67 (1): 210–229, doi:10.1016/0021-8693(80)90318-X, ISSN 0021-8693, MR 0595029
• Traverso, Carlo (1970), "Seminormality and Picard group", Ann. Scuola Norm. Sup. Pisa (3), 24: 585–595, MR 0277542
• Vitulli, Marie A. (2011), "Weak normality and seminormality" (PDF), Commutative algebra---Noetherian and non-Noetherian perspectives, Berlin, New York: Springer-Verlag, pp. 441–480, arXiv:0906.3334, doi:10.1007/978-1-4419-6990-3_17, ISBN 978-1-4419-6989-7, MR 2762521
• Charles Weibel, The K-book: An introduction to algebraic K-theory