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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067600/n0676001.png" /> be a commutative ring with identity and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067600/n0676002.png" /> a commutative ring containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067600/n0676003.png" />, with the same identity element. An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067600/n0676004.png" /> is integral over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067600/n0676005.png" /> if there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067600/n0676006.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067600/n0676007.png" />. The integral closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067600/n0676008.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067600/n0676009.png" /> is the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067600/n06760010.png" /> which are integral over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067600/n06760011.png" />. It is a subring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067600/n06760012.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067600/n06760013.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067600/n06760014.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067600/n06760015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067600/n06760016.png" /> is said to be integrally closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067600/n06760017.png" /> (cf. also [[Integral ring|Integral 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|Integral ring]]).
  
A commutative ring with identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067600/n06760018.png" /> is called normal if it is reduced (i.e. has no nilpotents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067600/n06760019.png" />) and is integrally closed in its complete ring of fractions (cf. [[Localization in a commutative algebra|Localization in a commutative algebra]]). Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067600/n06760020.png" /> is normal if for each prime ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067600/n06760021.png" /> the localization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067600/n06760022.png" /> is an [[Integral domain|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.
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A commutative ring with identity $R$ is called normal if it is reduced (i.e. has no nilpotents $\neq 0$) 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|Noetherian ring]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067600/n06760023.png" /> is normal if and only if it satisfies the two conditions: i) for every prime ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067600/n06760024.png" /> of height 1, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067600/n06760025.png" /> is regular (and hence a discrete valuation ring); and ii) for every prime ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067600/n06760026.png" /> of height <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067600/n06760027.png" /> the depth (cf. also [[Depth of a module|Depth of a module]]) is also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067600/n06760028.png" />. (Cf. [[#References|[a3]]], p. 125.)
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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|Depth of a module]]) is also $\ge 2$. (Cf. [[#References|[a3]]], p. 125.)
  
 
====References====
 
====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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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Nagata,  "Local rings" , Interscience  (1962)  pp. Chapt. III, §23</TD></TR>
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Matsumura,  "Commutative algebra" , Benjamin  (1970)</TD></TR>
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</table>
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Revision as of 21:19, 8 November 2014

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 nilpotents $\neq 0$) 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.)

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)
How to Cite This Entry:
Normal ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_ring&oldid=34377