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Let $R$ be a [[commutative ring]] [[unital ring|with identity]] and $S$ a commutative ring containing $R$, with the same identity element. An element $s \in S$ is integral over $R$ if there are $c_i \in R$ such that $s^n + c_1s^{n-1} + \cdots + c_n = 0$. The integral closure of $R$ in $S$ is the set of all $s \in S$ which are integral over $R$. It is a subring $\bar R$ of $S$ containing $R$. If $\bar R = R$, then $R$ is said to be integrally closed in $S$ (cf. also [[Integral ring]]).

A commutative ring with identity $R$ is called normal if it is reduced (i.e. has no [[nilpotent element]]s $\neq 0$) and is integrally closed in its complete ring of fractions (cf. [[Localization in a commutative algebra]]). Thus, $R$ is normal if for each prime ideal $\mathfrak{p}$ the localization $R_{\mathfrak{p}}$ 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]] $A$ is normal if and only if it satisfies the two conditions: i) for every prime ideal $\mathfrak{p}$ of height 1, $A_{\mathfrak{p}}$ is regular (and hence a discrete valuation ring); and ii) for every prime ideal $\mathfrak{p}$ of height $\ge 2$ the depth (cf. also [[Depth of a module]]) is also $\ge 2$. (Cf. [[#References|[a3]]], p. 125.)

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Nagata,   "Local rings" , Interscience  (1962)  pp. Chapt. III, §23</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Matsumura,   "Commutative algebra" , Benjamin  (1970)</TD></TR>
</table>

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[[Category:Associative rings and algebras]]