Seminormal ring
In algebra, a seminormal ring is a commutative reduced ring in which, whenever x, y satisfy [math]\displaystyle{ x^3 = y^2 }[/math], there is s with [math]\displaystyle{ s^2 = x }[/math] and [math]\displaystyle{ s^3 = y }[/math]. 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 [math]\displaystyle{ \mathbb{Z}[x, y]/xy }[/math], or the ring of a nodal curve.
In general, a reduced scheme [math]\displaystyle{ X }[/math] can be said to be seminormal if every morphism [math]\displaystyle{ Y \to X }[/math] 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
- Traverso, Carlo (1970), "Seminormality and Picard group", Ann. Scuola Norm. Sup. Pisa (3) 24: 585–595, http://www.numdam.org/item?id=ASNSP_1970_3_24_4_585_0
- Vitulli, Marie A. (2011), "Weak normality and seminormality", Commutative algebra---Noetherian and non-Noetherian perspectives, Berlin, New York: Springer-Verlag, pp. 441–480, doi:10.1007/978-1-4419-6990-3_17, ISBN 978-1-4419-6989-7, http://pages.uoregon.edu/vitulli/WeakAndSeminormality.pdf
- Charles Weibel, The K-book: An introduction to algebraic K-theory
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