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 {{harvtxt|Swan|1980}} as a simplification of the original definition of {{harvtxt|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 $\mathbb{Z}[x, y]/xy$ , or the ring of a nodal curve.

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

{{Citation | last1=Swan | first1=Richard G. | title=On seminormality | doi=10.1016/0021-8693(80)90318-X |mr=595029 | year=1980 | journal=Journal of Algebra | issn=0021-8693 | volume=67 | issue=1 | pages=210–229| doi-access=free }}
{{Citation | last1=Traverso | first1=Carlo | title=Seminormality and Picard group | url=http://www.numdam.org/item?id=ASNSP_1970_3_24_4_585_0 |mr=0277542 | year=1970 | journal=Ann. Scuola Norm. Sup. Pisa (3) | volume=24 | pages=585–595}}
{{Citation | last1=Vitulli | first1=Marie A.|authorlink= Marie A. Vitulli | title=Commutative algebra---Noetherian and non-Noetherian perspectives | chapter-url=http://pages.uoregon.edu/vitulli/WeakAndSeminormality.pdf | publisher=Springer-Verlag | location=Berlin, New York | doi=10.1007/978-1-4419-6990-3_17 |mr=2762521 | year=2011 | chapter=Weak normality and seminormality | pages=441–480| arxiv=0906.3334 | isbn=978-1-4419-6989-7}}
Charles Weibel, [http://www.math.rutgers.edu/~weibel/Kbook.html The K-book: An introduction to algebraic K-theory]

Category:Commutative algebra

Category:Ring theory