Difference between revisions of "Integral ideal"
From Encyclopedia of Mathematics
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| − | An [[Ideal|ideal]] of the field | + | {{TEX|done}} |
| + | An [[Ideal|ideal]] of the field $Q$ relative to a ring $A$ (here $Q$ is the field of fractions of $A$, cf. [[Fractions, ring of|Fractions, ring of]]) that lies entirely in $A$. An integral ideal is an ideal in $A$, and, conversely, every ideal of $A$ is an integral ideal of the field of fractions $Q$ of $A$. | ||
====Comments==== | ====Comments==== | ||
| − | An ideal of the field | + | An ideal of the field $Q$ relative to a ring $A\subset Q$ is an $A$-submodule of the $A$-module $Q$. These are also called fractional ideals, cf. [[Fractional ideal|Fractional ideal]]. |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Weiss, "Algebraic number theory" , McGraw-Hill (1963)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Weiss, "Algebraic number theory" , McGraw-Hill (1963)</TD></TR></table> | ||
Latest revision as of 11:51, 22 August 2014
An ideal of the field $Q$ relative to a ring $A$ (here $Q$ is the field of fractions of $A$, cf. Fractions, ring of) that lies entirely in $A$. An integral ideal is an ideal in $A$, and, conversely, every ideal of $A$ is an integral ideal of the field of fractions $Q$ of $A$.
Comments
An ideal of the field $Q$ relative to a ring $A\subset Q$ is an $A$-submodule of the $A$-module $Q$. These are also called fractional ideals, cf. Fractional ideal.
References
| [a1] | E. Weiss, "Algebraic number theory" , McGraw-Hill (1963) |
How to Cite This Entry:
Integral ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integral_ideal&oldid=33086
Integral ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integral_ideal&oldid=33086
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article