Normal ring
Let be a commutative ring with identity and
a commutative ring containing
, with the same identity element. An element
is integral over
if there are
such that
. The integral closure of
in
is the set of all
which are integral over
. It is a subring
of
containing
. If
,
is said to be integrally closed in
(cf. also Integral ring).
A commutative ring with identity is called normal if it is reduced (i.e. has no nilpotents
) and is integrally closed in its complete ring of fractions (cf. Localization in a commutative algebra). Thus,
is normal if for each prime ideal
the localization
is an integral domain and is closed in its field of fractions. In some of the literature a normal ring is also required to be an integral domain.
A Noetherian ring is normal if and only if it satisfies the two conditions: i) for every prime ideal
of height 1,
is regular (and hence a discrete valuation ring); and ii) for every prime ideal
of height
the depth (cf. also Depth of a module) is also
. (Cf. [a3], p. 125.)
References
[a1] | N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) |
[a2] | M. Nagata, "Local rings" , Interscience (1962) pp. Chapt. III, §23 |
[a3] | H. Matsumura, "Commutative algebra" , Benjamin (1970) |
Normal ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_ring&oldid=19257