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. [[ | + | 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> | + | <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> | ||
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
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