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| − | A subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041220/f0412201.png" /> of the field of fractions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041220/f0412202.png" /> of a commutative integral domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041220/f0412203.png" /> of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041220/f0412204.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041220/f0412205.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041220/f0412206.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041220/f0412207.png" /> is an ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041220/f0412208.png" />. In other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041220/f0412209.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041220/f04122010.png" />-submodule of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041220/f04122011.png" /> all elements of which permit a common denominator, i.e. there exists an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041220/f04122012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041220/f04122013.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041220/f04122014.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041220/f04122015.png" />. Fractional ideals form a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041220/f04122016.png" /> with unit element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041220/f04122017.png" /> with respect to multiplication. This semi-group is a group for Dedekind rings and only for such rings (cf. [[Dedekind ring|Dedekind ring]]). The invertible elements of the semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041220/f04122018.png" /> are said to be invertible ideals. Each invertible ideal has a finite basis over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041220/f04122019.png" />. | + | {{TEX|done}} |
| | + | A subset $Q$ of the field of fractions $K$ of a commutative integral domain $R$ of the form $Q=a^{-1}I$, where $a\in R$, $a\neq0$, and $I$ is an ideal of $R$. In other words, $Q$ is an $R$-submodule of the field $K$ all elements of which permit a common denominator, i.e. there exists an element $a\in R$, $a\neq0$, such that $ax\in R$ for all $x\in Q$. Fractional ideals form a semi-group $\mathfrak A$ with unit element $R$ with respect to multiplication. This semi-group is a group for Dedekind rings and only for such rings (cf. [[Dedekind ring|Dedekind ring]]). The invertible elements of the semi-group $\mathfrak A$ are said to be invertible ideals. Each invertible ideal has a finite basis over $R$. |
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Revision as of 11:49, 12 April 2014
A subset $Q$ of the field of fractions $K$ of a commutative integral domain $R$ of the form $Q=a^{-1}I$, where $a\in R$, $a\neq0$, and $I$ is an ideal of $R$. In other words, $Q$ is an $R$-submodule of the field $K$ all elements of which permit a common denominator, i.e. there exists an element $a\in R$, $a\neq0$, such that $ax\in R$ for all $x\in Q$. Fractional ideals form a semi-group $\mathfrak A$ with unit element $R$ with respect to multiplication. This semi-group is a group for Dedekind rings and only for such rings (cf. Dedekind ring). The invertible elements of the semi-group $\mathfrak A$ are said to be invertible ideals. Each invertible ideal has a finite basis over $R$.
References
| [1] | O. Zariski, P. Samuel, "Commutative algebra" , 1 , Springer (1975) |
| [2] | N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) |
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How to Cite This Entry:
Fractional ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fractional_ideal&oldid=31616
This article was adapted from an original article by L.A. Bokut (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article