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Difference between revisions of "Integral ideal"

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An [[Ideal|ideal]] of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051500/i0515001.png" /> relative to a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051500/i0515002.png" /> (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051500/i0515003.png" /> is the field of fractions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051500/i0515004.png" />, cf. [[Fractions, ring of|Fractions, ring of]]) that lies entirely in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051500/i0515005.png" />. An integral ideal is an ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051500/i0515006.png" />, and, conversely, every ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051500/i0515007.png" /> is an integral ideal of the field of fractions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051500/i0515008.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051500/i0515009.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051500/i05150010.png" /> relative to a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051500/i05150011.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051500/i05150012.png" />-submodule of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051500/i05150013.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051500/i05150014.png" />. These are also called fractional ideals, cf. [[Fractional ideal|Fractional ideal]].
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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
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article