In algebra, a seminormal ring is a commutative reduced ring in which, whenever x, y satisfy , there is s with and . 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 , or the ring of a nodel curve.
In general, a reduced scheme can be said to be seminormal if every morphism 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.
- Swan, Richard G. (1980), "On seminormality", Journal of Algebra, 67 (1): 210–229, ISSN 0021-8693, MR 595029, doi:10.1016/0021-8693(80)90318-X
- 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", Commutative algebra---Noetherian and non-Noetherian perspectives (PDF), Berlin, New York: Springer-Verlag, pp. 441–480, MR 2762521, doi:10.1007/978-1-4419-6990-3_17
- Charles Weibel, The K-book: An introduction to algebraic K-theory
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