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Difference between revisions of "Fractional ideal"

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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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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====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  O. Zariski,  P. Samuel,  "Commutative algebra" , '''1''' , Springer  (1975)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR></table>
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<table>
 
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<TR><TD valign="top">[1]</TD> <TD valign="top">  O. Zariski,  P. Samuel,  "Commutative algebra" , '''1''' , Springer  (1975)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR>
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</table>

Revision as of 20:48, 28 November 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)
How to Cite This Entry:
Fractional ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fractional_ideal&oldid=35043
This article was adapted from an original article by L.A. Bokut (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article