What does it mean to say a quotient of fractional ideals?

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Let $mathbbK$ be an algebraic number field, and $R$ its ring of integers. If $I$ and $J$ are two fractional ideal in $mathbbK$ such that $J subseteq I$, what does it mean to say $I/J$? What is the difference from integral ideals?




abstract-algebra ring-theory algebraic-number-theory ideals




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    Maybe it is meant $;Icdot J^-1;$ ? Do you know in some cases the set of all fractional ideals (done with an integer domain, field of fractions and etc.) is a group?
    – DonAntonio
    Jul 21 at 19:42










  • Yes I know that, but is this equivalent to say $I$ modulo $J$?
    – C.S.
    Jul 21 at 21:36










  • $I/J$ is a torsion module over dedekind ring $O_K$. It is not even an integral ideal. Note that $J$ contains a $Q$ basis of $K$. All integral ideals are torsion free as submodule of $O_K$. The text might mean $IJ^-1$ which is another fractional ideal. Once you multiply $IJ^-1$ by an appropriate factor of $K$, it will become integral ideal.
    – user45765
    Jul 21 at 22:19














up vote
0
down vote

favorite












Let $mathbbK$ be an algebraic number field, and $R$ its ring of integers. If $I$ and $J$ are two fractional ideal in $mathbbK$ such that $J subseteq I$, what does it mean to say $I/J$? What is the difference from integral ideals?




abstract-algebra ring-theory algebraic-number-theory ideals




share|cite|improve this question

















  • 1




    Maybe it is meant $;Icdot J^-1;$ ? Do you know in some cases the set of all fractional ideals (done with an integer domain, field of fractions and etc.) is a group?
    – DonAntonio
    Jul 21 at 19:42










  • Yes I know that, but is this equivalent to say $I$ modulo $J$?
    – C.S.
    Jul 21 at 21:36










  • $I/J$ is a torsion module over dedekind ring $O_K$. It is not even an integral ideal. Note that $J$ contains a $Q$ basis of $K$. All integral ideals are torsion free as submodule of $O_K$. The text might mean $IJ^-1$ which is another fractional ideal. Once you multiply $IJ^-1$ by an appropriate factor of $K$, it will become integral ideal.
    – user45765
    Jul 21 at 22:19












up vote
0
down vote

favorite









up vote
0
down vote

favorite











Let $mathbbK$ be an algebraic number field, and $R$ its ring of integers. If $I$ and $J$ are two fractional ideal in $mathbbK$ such that $J subseteq I$, what does it mean to say $I/J$? What is the difference from integral ideals?




abstract-algebra ring-theory algebraic-number-theory ideals




share|cite|improve this question













Let $mathbbK$ be an algebraic number field, and $R$ its ring of integers. If $I$ and $J$ are two fractional ideal in $mathbbK$ such that $J subseteq I$, what does it mean to say $I/J$? What is the difference from integral ideals?





abstract-algebra ring-theory algebraic-number-theory ideals





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Jul 21 at 19:48
























asked Jul 21 at 19:39









C.S.

134




134







  • 1




    Maybe it is meant $;Icdot J^-1;$ ? Do you know in some cases the set of all fractional ideals (done with an integer domain, field of fractions and etc.) is a group?
    – DonAntonio
    Jul 21 at 19:42










  • Yes I know that, but is this equivalent to say $I$ modulo $J$?
    – C.S.
    Jul 21 at 21:36










  • $I/J$ is a torsion module over dedekind ring $O_K$. It is not even an integral ideal. Note that $J$ contains a $Q$ basis of $K$. All integral ideals are torsion free as submodule of $O_K$. The text might mean $IJ^-1$ which is another fractional ideal. Once you multiply $IJ^-1$ by an appropriate factor of $K$, it will become integral ideal.
    – user45765
    Jul 21 at 22:19












  • 1




    Maybe it is meant $;Icdot J^-1;$ ? Do you know in some cases the set of all fractional ideals (done with an integer domain, field of fractions and etc.) is a group?
    – DonAntonio
    Jul 21 at 19:42










  • Yes I know that, but is this equivalent to say $I$ modulo $J$?
    – C.S.
    Jul 21 at 21:36










  • $I/J$ is a torsion module over dedekind ring $O_K$. It is not even an integral ideal. Note that $J$ contains a $Q$ basis of $K$. All integral ideals are torsion free as submodule of $O_K$. The text might mean $IJ^-1$ which is another fractional ideal. Once you multiply $IJ^-1$ by an appropriate factor of $K$, it will become integral ideal.
    – user45765
    Jul 21 at 22:19







1




1




Maybe it is meant $;Icdot J^-1;$ ? Do you know in some cases the set of all fractional ideals (done with an integer domain, field of fractions and etc.) is a group?
– DonAntonio
Jul 21 at 19:42




Maybe it is meant $;Icdot J^-1;$ ? Do you know in some cases the set of all fractional ideals (done with an integer domain, field of fractions and etc.) is a group?
– DonAntonio
Jul 21 at 19:42












Yes I know that, but is this equivalent to say $I$ modulo $J$?
– C.S.
Jul 21 at 21:36




Yes I know that, but is this equivalent to say $I$ modulo $J$?
– C.S.
Jul 21 at 21:36












$I/J$ is a torsion module over dedekind ring $O_K$. It is not even an integral ideal. Note that $J$ contains a $Q$ basis of $K$. All integral ideals are torsion free as submodule of $O_K$. The text might mean $IJ^-1$ which is another fractional ideal. Once you multiply $IJ^-1$ by an appropriate factor of $K$, it will become integral ideal.
– user45765
Jul 21 at 22:19




$I/J$ is a torsion module over dedekind ring $O_K$. It is not even an integral ideal. Note that $J$ contains a $Q$ basis of $K$. All integral ideals are torsion free as submodule of $O_K$. The text might mean $IJ^-1$ which is another fractional ideal. Once you multiply $IJ^-1$ by an appropriate factor of $K$, it will become integral ideal.
– user45765
Jul 21 at 22:19















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