Notion of 2-sided artinian ring?












1












$begingroup$


An Artinian ring is defined to be a ring that is both left Artinian and right Artinian. So satisfies Descending chain condition (DCC) on left and right ideals. This implies it also satisfies DCC for the set of 2-sided ideals.



Edit: But for the converse, having DCC on 2-sided ideals give DCC on left (or right) ideals, seems nontrivial to me. Since there can be many more left ideals than 2-sided ideals..



So is there a notion of "2-sided Artinian" with the DCC condition only on 2sided ideals?



I can't find any reference to it or what it would be used for.






reference-request ring-theory noncommutative-algebra







share|cite|improve this question











$endgroup$












  • $begingroup$
    @LukasKofler I've edited my question. If the converse is true I don't know why
    $endgroup$
    – Ted Jh
    Dec 26 '18 at 13:09
















1












$begingroup$


An Artinian ring is defined to be a ring that is both left Artinian and right Artinian. So satisfies Descending chain condition (DCC) on left and right ideals. This implies it also satisfies DCC for the set of 2-sided ideals.



Edit: But for the converse, having DCC on 2-sided ideals give DCC on left (or right) ideals, seems nontrivial to me. Since there can be many more left ideals than 2-sided ideals..



So is there a notion of "2-sided Artinian" with the DCC condition only on 2sided ideals?



I can't find any reference to it or what it would be used for.






reference-request ring-theory noncommutative-algebra







share|cite|improve this question











$endgroup$












  • $begingroup$
    @LukasKofler I've edited my question. If the converse is true I don't know why
    $endgroup$
    – Ted Jh
    Dec 26 '18 at 13:09














1












1








1





$begingroup$


An Artinian ring is defined to be a ring that is both left Artinian and right Artinian. So satisfies Descending chain condition (DCC) on left and right ideals. This implies it also satisfies DCC for the set of 2-sided ideals.



Edit: But for the converse, having DCC on 2-sided ideals give DCC on left (or right) ideals, seems nontrivial to me. Since there can be many more left ideals than 2-sided ideals..



So is there a notion of "2-sided Artinian" with the DCC condition only on 2sided ideals?



I can't find any reference to it or what it would be used for.






reference-request ring-theory noncommutative-algebra







share|cite|improve this question











$endgroup$




An Artinian ring is defined to be a ring that is both left Artinian and right Artinian. So satisfies Descending chain condition (DCC) on left and right ideals. This implies it also satisfies DCC for the set of 2-sided ideals.



Edit: But for the converse, having DCC on 2-sided ideals give DCC on left (or right) ideals, seems nontrivial to me. Since there can be many more left ideals than 2-sided ideals..



So is there a notion of "2-sided Artinian" with the DCC condition only on 2sided ideals?



I can't find any reference to it or what it would be used for.






reference-request ring-theory noncommutative-algebra




reference-request ring-theory noncommutative-algebra






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 26 '18 at 14:39







Ted Jh

















asked Dec 26 '18 at 11:33









Ted JhTed Jh

644




644












  • $begingroup$
    @LukasKofler I've edited my question. If the converse is true I don't know why
    $endgroup$
    – Ted Jh
    Dec 26 '18 at 13:09


















  • $begingroup$
    @LukasKofler I've edited my question. If the converse is true I don't know why
    $endgroup$
    – Ted Jh
    Dec 26 '18 at 13:09
















$begingroup$
@LukasKofler I've edited my question. If the converse is true I don't know why
$endgroup$
– Ted Jh
Dec 26 '18 at 13:09




$begingroup$
@LukasKofler I've edited my question. If the converse is true I don't know why
$endgroup$
– Ted Jh
Dec 26 '18 at 13:09










2 Answers
2






active

oldest

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1












$begingroup$


So is there a notion of "2-sided Artinian" with the DCC condition only on 2sided ideals?




I think the typical thing to do is to say "DCC on two-sided ideals," without using the "Artinian" adjective. I see lots of hits for this sort of thing when I search.




But for the converse, having DCC on 2-sided ideals give DCC on left (or right) ideals, seems nontrivial to me. Since there can be many more left ideals than 2-sided ideals..




Of course, your intuition is right it is not true that DCC on two-sided ideals would imply DCC on left ideals. For example, you can take any non-Artinian (or non-Noetherian) simple ring. It would necessarily have only two ideals, but infinite descending (resp. ascending and descending!) chains of left ideals.






share|cite|improve this answer









$endgroup$





















    0












    $begingroup$

    I believe my confusion was in the terminology of the definition: where it says "descending chain condition on left and right ideals" it means DCC on 2-sided ideals, rather than DCC on all left ideals and DCC on all right ideals...?






    share|cite|improve this answer









    $endgroup$









    • 1




      $begingroup$
      It does mean DCC on left ideals and on right ideals
      $endgroup$
      – rschwieb
      Dec 26 '18 at 16:32












    • $begingroup$
      @rschwieb thanks for clarifying.
      $endgroup$
      – Ted Jh
      Dec 26 '18 at 18:02











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    2 Answers
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    2 Answers
    2






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    active

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    1












    $begingroup$


    So is there a notion of "2-sided Artinian" with the DCC condition only on 2sided ideals?




    I think the typical thing to do is to say "DCC on two-sided ideals," without using the "Artinian" adjective. I see lots of hits for this sort of thing when I search.




    But for the converse, having DCC on 2-sided ideals give DCC on left (or right) ideals, seems nontrivial to me. Since there can be many more left ideals than 2-sided ideals..




    Of course, your intuition is right it is not true that DCC on two-sided ideals would imply DCC on left ideals. For example, you can take any non-Artinian (or non-Noetherian) simple ring. It would necessarily have only two ideals, but infinite descending (resp. ascending and descending!) chains of left ideals.






    share|cite|improve this answer









    $endgroup$


















      1












      $begingroup$


      So is there a notion of "2-sided Artinian" with the DCC condition only on 2sided ideals?




      I think the typical thing to do is to say "DCC on two-sided ideals," without using the "Artinian" adjective. I see lots of hits for this sort of thing when I search.




      But for the converse, having DCC on 2-sided ideals give DCC on left (or right) ideals, seems nontrivial to me. Since there can be many more left ideals than 2-sided ideals..




      Of course, your intuition is right it is not true that DCC on two-sided ideals would imply DCC on left ideals. For example, you can take any non-Artinian (or non-Noetherian) simple ring. It would necessarily have only two ideals, but infinite descending (resp. ascending and descending!) chains of left ideals.






      share|cite|improve this answer









      $endgroup$
















        1












        1








        1





        $begingroup$


        So is there a notion of "2-sided Artinian" with the DCC condition only on 2sided ideals?




        I think the typical thing to do is to say "DCC on two-sided ideals," without using the "Artinian" adjective. I see lots of hits for this sort of thing when I search.




        But for the converse, having DCC on 2-sided ideals give DCC on left (or right) ideals, seems nontrivial to me. Since there can be many more left ideals than 2-sided ideals..




        Of course, your intuition is right it is not true that DCC on two-sided ideals would imply DCC on left ideals. For example, you can take any non-Artinian (or non-Noetherian) simple ring. It would necessarily have only two ideals, but infinite descending (resp. ascending and descending!) chains of left ideals.






        share|cite|improve this answer









        $endgroup$




        So is there a notion of "2-sided Artinian" with the DCC condition only on 2sided ideals?




        I think the typical thing to do is to say "DCC on two-sided ideals," without using the "Artinian" adjective. I see lots of hits for this sort of thing when I search.




        But for the converse, having DCC on 2-sided ideals give DCC on left (or right) ideals, seems nontrivial to me. Since there can be many more left ideals than 2-sided ideals..




        Of course, your intuition is right it is not true that DCC on two-sided ideals would imply DCC on left ideals. For example, you can take any non-Artinian (or non-Noetherian) simple ring. It would necessarily have only two ideals, but infinite descending (resp. ascending and descending!) chains of left ideals.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 26 '18 at 15:07









        rschwiebrschwieb

        106k12102250




        106k12102250























            0












            $begingroup$

            I believe my confusion was in the terminology of the definition: where it says "descending chain condition on left and right ideals" it means DCC on 2-sided ideals, rather than DCC on all left ideals and DCC on all right ideals...?






            share|cite|improve this answer









            $endgroup$









            • 1




              $begingroup$
              It does mean DCC on left ideals and on right ideals
              $endgroup$
              – rschwieb
              Dec 26 '18 at 16:32












            • $begingroup$
              @rschwieb thanks for clarifying.
              $endgroup$
              – Ted Jh
              Dec 26 '18 at 18:02
















            0












            $begingroup$

            I believe my confusion was in the terminology of the definition: where it says "descending chain condition on left and right ideals" it means DCC on 2-sided ideals, rather than DCC on all left ideals and DCC on all right ideals...?






            share|cite|improve this answer









            $endgroup$









            • 1




              $begingroup$
              It does mean DCC on left ideals and on right ideals
              $endgroup$
              – rschwieb
              Dec 26 '18 at 16:32












            • $begingroup$
              @rschwieb thanks for clarifying.
              $endgroup$
              – Ted Jh
              Dec 26 '18 at 18:02














            0












            0








            0





            $begingroup$

            I believe my confusion was in the terminology of the definition: where it says "descending chain condition on left and right ideals" it means DCC on 2-sided ideals, rather than DCC on all left ideals and DCC on all right ideals...?






            share|cite|improve this answer









            $endgroup$



            I believe my confusion was in the terminology of the definition: where it says "descending chain condition on left and right ideals" it means DCC on 2-sided ideals, rather than DCC on all left ideals and DCC on all right ideals...?







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Dec 26 '18 at 15:01









            Ted JhTed Jh

            644




            644








            • 1




              $begingroup$
              It does mean DCC on left ideals and on right ideals
              $endgroup$
              – rschwieb
              Dec 26 '18 at 16:32












            • $begingroup$
              @rschwieb thanks for clarifying.
              $endgroup$
              – Ted Jh
              Dec 26 '18 at 18:02














            • 1




              $begingroup$
              It does mean DCC on left ideals and on right ideals
              $endgroup$
              – rschwieb
              Dec 26 '18 at 16:32












            • $begingroup$
              @rschwieb thanks for clarifying.
              $endgroup$
              – Ted Jh
              Dec 26 '18 at 18:02








            1




            1




            $begingroup$
            It does mean DCC on left ideals and on right ideals
            $endgroup$
            – rschwieb
            Dec 26 '18 at 16:32






            $begingroup$
            It does mean DCC on left ideals and on right ideals
            $endgroup$
            – rschwieb
            Dec 26 '18 at 16:32














            $begingroup$
            @rschwieb thanks for clarifying.
            $endgroup$
            – Ted Jh
            Dec 26 '18 at 18:02




            $begingroup$
            @rschwieb thanks for clarifying.
            $endgroup$
            – Ted Jh
            Dec 26 '18 at 18:02


















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