Normal ring

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Let $R$ be a commutative 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 elements $\neq 0$) 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. [a3], p. 125.)


[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)
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