Additive group of local rings Announcing the arrival of Valued Associate #679: Cesar Manara Unicorn Meta Zoo #1: Why another podcast?Krull's intersection theorem for commutative local not necessarily noetherian ringsIs the class of additive groups of rings axiomatizable?Characterization of non-commutative local rings of orders 64 and 128Local rings with simple radicalA question on local ringsIs every commutative group structure underlying at least one (unitary, commutative) ring structureProjecting solutions of Hermitian forms over local ringsautomorphisms of local rings vs local change of coordinatesQuotients of rings with finite free additive groupWhen is a zero dimensional local ring a chain ring?

Additive group of local rings



Announcing the arrival of Valued Associate #679: Cesar Manara
Unicorn Meta Zoo #1: Why another podcast?Krull's intersection theorem for commutative local not necessarily noetherian ringsIs the class of additive groups of rings axiomatizable?Characterization of non-commutative local rings of orders 64 and 128Local rings with simple radicalA question on local ringsIs every commutative group structure underlying at least one (unitary, commutative) ring structureProjecting solutions of Hermitian forms over local ringsautomorphisms of local rings vs local change of coordinatesQuotients of rings with finite free additive groupWhen is a zero dimensional local ring a chain ring?










5












$begingroup$


Is there a theory or characterization for those finite $p$-groups that can be considered as the additive group of a finite local commutative ring with identity?










share|cite|improve this question









$endgroup$
















    5












    $begingroup$


    Is there a theory or characterization for those finite $p$-groups that can be considered as the additive group of a finite local commutative ring with identity?










    share|cite|improve this question









    $endgroup$














      5












      5








      5





      $begingroup$


      Is there a theory or characterization for those finite $p$-groups that can be considered as the additive group of a finite local commutative ring with identity?










      share|cite|improve this question









      $endgroup$




      Is there a theory or characterization for those finite $p$-groups that can be considered as the additive group of a finite local commutative ring with identity?







      ac.commutative-algebra ra.rings-and-algebras abelian-groups local-rings






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked 8 hours ago









      Lisa_KLisa_K

      654




      654




















          1 Answer
          1






          active

          oldest

          votes


















          7












          $begingroup$

          Any nonzero finite abelian $ p $ - group is the additive group of a commutative local ring.



          Proof : Let $ G $ be such a group, by the structure theorem you can write it as $ mathbbZ/p^k mathbbZ times M $ where $ p^k = exp (G) $. Then $ M $ naturally has the structure of a $ mathbbZ/p^k mathbbZ $ - module.



          Now define a multiplication on $ G $ by $ (x, m)(y, n) := (xy, ym + xn) $. Clearly $ (1,0) $ is a unity.



          Now let's prove it's local with maximal ideal $ (x,m) mid p $ divides $x $.



          Indeed clearly this is a proper ideal, and now if $ p $ doesn't divide $ x $ then $ x $ is invertible modulo $ p^k $, let $ y $ be its inverse. Then $(x,m) (y, -y^2m) = (xy, -xy^2m + ym) = (1,0) $ so $ (x,m) $ is invertible : therefore the complement of our ideal is the set of nonunits, which implies that our ring is local.



          This example can be generalized of course : whenever $ R $ is a ring, $ M $ an $ R $ - module, you can "adjoin" $ M $ as an ideal to $ R $, this is where my construction comes from.






          share|cite|improve this answer











          $endgroup$








          • 2




            $begingroup$
            Precisely every nonzero such group.
            $endgroup$
            – YCor
            7 hours ago










          • $begingroup$
            @YCor : indeed, let me correct that
            $endgroup$
            – Max
            7 hours ago










          • $begingroup$
            What is the $M$ in $G = mathbb Z/p^kmathbb Z times M$?
            $endgroup$
            – LSpice
            6 hours ago






          • 2




            $begingroup$
            @LSpice : you can see it as $G/(mathbbZ/p^kmathbbZ)$ for instance; it comes from the structure theorem for finite abelian groups
            $endgroup$
            – Max
            6 hours ago











          Your Answer








          StackExchange.ready(function()
          var channelOptions =
          tags: "".split(" "),
          id: "504"
          ;
          initTagRenderer("".split(" "), "".split(" "), channelOptions);

          StackExchange.using("externalEditor", function()
          // Have to fire editor after snippets, if snippets enabled
          if (StackExchange.settings.snippets.snippetsEnabled)
          StackExchange.using("snippets", function()
          createEditor();
          );

          else
          createEditor();

          );

          function createEditor()
          StackExchange.prepareEditor(
          heartbeatType: 'answer',
          autoActivateHeartbeat: false,
          convertImagesToLinks: true,
          noModals: true,
          showLowRepImageUploadWarning: true,
          reputationToPostImages: 10,
          bindNavPrevention: true,
          postfix: "",
          imageUploader:
          brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
          contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
          allowUrls: true
          ,
          noCode: true, onDemand: true,
          discardSelector: ".discard-answer"
          ,immediatelyShowMarkdownHelp:true
          );



          );













          draft saved

          draft discarded


















          StackExchange.ready(
          function ()
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f329877%2fadditive-group-of-local-rings%23new-answer', 'question_page');

          );

          Post as a guest















          Required, but never shown

























          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          7












          $begingroup$

          Any nonzero finite abelian $ p $ - group is the additive group of a commutative local ring.



          Proof : Let $ G $ be such a group, by the structure theorem you can write it as $ mathbbZ/p^k mathbbZ times M $ where $ p^k = exp (G) $. Then $ M $ naturally has the structure of a $ mathbbZ/p^k mathbbZ $ - module.



          Now define a multiplication on $ G $ by $ (x, m)(y, n) := (xy, ym + xn) $. Clearly $ (1,0) $ is a unity.



          Now let's prove it's local with maximal ideal $ (x,m) mid p $ divides $x $.



          Indeed clearly this is a proper ideal, and now if $ p $ doesn't divide $ x $ then $ x $ is invertible modulo $ p^k $, let $ y $ be its inverse. Then $(x,m) (y, -y^2m) = (xy, -xy^2m + ym) = (1,0) $ so $ (x,m) $ is invertible : therefore the complement of our ideal is the set of nonunits, which implies that our ring is local.



          This example can be generalized of course : whenever $ R $ is a ring, $ M $ an $ R $ - module, you can "adjoin" $ M $ as an ideal to $ R $, this is where my construction comes from.






          share|cite|improve this answer











          $endgroup$








          • 2




            $begingroup$
            Precisely every nonzero such group.
            $endgroup$
            – YCor
            7 hours ago










          • $begingroup$
            @YCor : indeed, let me correct that
            $endgroup$
            – Max
            7 hours ago










          • $begingroup$
            What is the $M$ in $G = mathbb Z/p^kmathbb Z times M$?
            $endgroup$
            – LSpice
            6 hours ago






          • 2




            $begingroup$
            @LSpice : you can see it as $G/(mathbbZ/p^kmathbbZ)$ for instance; it comes from the structure theorem for finite abelian groups
            $endgroup$
            – Max
            6 hours ago















          7












          $begingroup$

          Any nonzero finite abelian $ p $ - group is the additive group of a commutative local ring.



          Proof : Let $ G $ be such a group, by the structure theorem you can write it as $ mathbbZ/p^k mathbbZ times M $ where $ p^k = exp (G) $. Then $ M $ naturally has the structure of a $ mathbbZ/p^k mathbbZ $ - module.



          Now define a multiplication on $ G $ by $ (x, m)(y, n) := (xy, ym + xn) $. Clearly $ (1,0) $ is a unity.



          Now let's prove it's local with maximal ideal $ (x,m) mid p $ divides $x $.



          Indeed clearly this is a proper ideal, and now if $ p $ doesn't divide $ x $ then $ x $ is invertible modulo $ p^k $, let $ y $ be its inverse. Then $(x,m) (y, -y^2m) = (xy, -xy^2m + ym) = (1,0) $ so $ (x,m) $ is invertible : therefore the complement of our ideal is the set of nonunits, which implies that our ring is local.



          This example can be generalized of course : whenever $ R $ is a ring, $ M $ an $ R $ - module, you can "adjoin" $ M $ as an ideal to $ R $, this is where my construction comes from.






          share|cite|improve this answer











          $endgroup$








          • 2




            $begingroup$
            Precisely every nonzero such group.
            $endgroup$
            – YCor
            7 hours ago










          • $begingroup$
            @YCor : indeed, let me correct that
            $endgroup$
            – Max
            7 hours ago










          • $begingroup$
            What is the $M$ in $G = mathbb Z/p^kmathbb Z times M$?
            $endgroup$
            – LSpice
            6 hours ago






          • 2




            $begingroup$
            @LSpice : you can see it as $G/(mathbbZ/p^kmathbbZ)$ for instance; it comes from the structure theorem for finite abelian groups
            $endgroup$
            – Max
            6 hours ago













          7












          7








          7





          $begingroup$

          Any nonzero finite abelian $ p $ - group is the additive group of a commutative local ring.



          Proof : Let $ G $ be such a group, by the structure theorem you can write it as $ mathbbZ/p^k mathbbZ times M $ where $ p^k = exp (G) $. Then $ M $ naturally has the structure of a $ mathbbZ/p^k mathbbZ $ - module.



          Now define a multiplication on $ G $ by $ (x, m)(y, n) := (xy, ym + xn) $. Clearly $ (1,0) $ is a unity.



          Now let's prove it's local with maximal ideal $ (x,m) mid p $ divides $x $.



          Indeed clearly this is a proper ideal, and now if $ p $ doesn't divide $ x $ then $ x $ is invertible modulo $ p^k $, let $ y $ be its inverse. Then $(x,m) (y, -y^2m) = (xy, -xy^2m + ym) = (1,0) $ so $ (x,m) $ is invertible : therefore the complement of our ideal is the set of nonunits, which implies that our ring is local.



          This example can be generalized of course : whenever $ R $ is a ring, $ M $ an $ R $ - module, you can "adjoin" $ M $ as an ideal to $ R $, this is where my construction comes from.






          share|cite|improve this answer











          $endgroup$



          Any nonzero finite abelian $ p $ - group is the additive group of a commutative local ring.



          Proof : Let $ G $ be such a group, by the structure theorem you can write it as $ mathbbZ/p^k mathbbZ times M $ where $ p^k = exp (G) $. Then $ M $ naturally has the structure of a $ mathbbZ/p^k mathbbZ $ - module.



          Now define a multiplication on $ G $ by $ (x, m)(y, n) := (xy, ym + xn) $. Clearly $ (1,0) $ is a unity.



          Now let's prove it's local with maximal ideal $ (x,m) mid p $ divides $x $.



          Indeed clearly this is a proper ideal, and now if $ p $ doesn't divide $ x $ then $ x $ is invertible modulo $ p^k $, let $ y $ be its inverse. Then $(x,m) (y, -y^2m) = (xy, -xy^2m + ym) = (1,0) $ so $ (x,m) $ is invertible : therefore the complement of our ideal is the set of nonunits, which implies that our ring is local.



          This example can be generalized of course : whenever $ R $ is a ring, $ M $ an $ R $ - module, you can "adjoin" $ M $ as an ideal to $ R $, this is where my construction comes from.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited 7 hours ago

























          answered 7 hours ago









          MaxMax

          6391619




          6391619







          • 2




            $begingroup$
            Precisely every nonzero such group.
            $endgroup$
            – YCor
            7 hours ago










          • $begingroup$
            @YCor : indeed, let me correct that
            $endgroup$
            – Max
            7 hours ago










          • $begingroup$
            What is the $M$ in $G = mathbb Z/p^kmathbb Z times M$?
            $endgroup$
            – LSpice
            6 hours ago






          • 2




            $begingroup$
            @LSpice : you can see it as $G/(mathbbZ/p^kmathbbZ)$ for instance; it comes from the structure theorem for finite abelian groups
            $endgroup$
            – Max
            6 hours ago












          • 2




            $begingroup$
            Precisely every nonzero such group.
            $endgroup$
            – YCor
            7 hours ago










          • $begingroup$
            @YCor : indeed, let me correct that
            $endgroup$
            – Max
            7 hours ago










          • $begingroup$
            What is the $M$ in $G = mathbb Z/p^kmathbb Z times M$?
            $endgroup$
            – LSpice
            6 hours ago






          • 2




            $begingroup$
            @LSpice : you can see it as $G/(mathbbZ/p^kmathbbZ)$ for instance; it comes from the structure theorem for finite abelian groups
            $endgroup$
            – Max
            6 hours ago







          2




          2




          $begingroup$
          Precisely every nonzero such group.
          $endgroup$
          – YCor
          7 hours ago




          $begingroup$
          Precisely every nonzero such group.
          $endgroup$
          – YCor
          7 hours ago












          $begingroup$
          @YCor : indeed, let me correct that
          $endgroup$
          – Max
          7 hours ago




          $begingroup$
          @YCor : indeed, let me correct that
          $endgroup$
          – Max
          7 hours ago












          $begingroup$
          What is the $M$ in $G = mathbb Z/p^kmathbb Z times M$?
          $endgroup$
          – LSpice
          6 hours ago




          $begingroup$
          What is the $M$ in $G = mathbb Z/p^kmathbb Z times M$?
          $endgroup$
          – LSpice
          6 hours ago




          2




          2




          $begingroup$
          @LSpice : you can see it as $G/(mathbbZ/p^kmathbbZ)$ for instance; it comes from the structure theorem for finite abelian groups
          $endgroup$
          – Max
          6 hours ago




          $begingroup$
          @LSpice : you can see it as $G/(mathbbZ/p^kmathbbZ)$ for instance; it comes from the structure theorem for finite abelian groups
          $endgroup$
          – Max
          6 hours ago

















          draft saved

          draft discarded
















































          Thanks for contributing an answer to MathOverflow!


          • Please be sure to answer the question. Provide details and share your research!

          But avoid


          • Asking for help, clarification, or responding to other answers.

          • Making statements based on opinion; back them up with references or personal experience.

          Use MathJax to format equations. MathJax reference.


          To learn more, see our tips on writing great answers.




          draft saved


          draft discarded














          StackExchange.ready(
          function ()
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f329877%2fadditive-group-of-local-rings%23new-answer', 'question_page');

          );

          Post as a guest















          Required, but never shown





















































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown

































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown







          Popular posts from this blog

          How to create a command for the “strange m” symbol in latex? Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)How do you make your own symbol when Detexify fails?Writing bold small caps with mathpazo packageplus-minus symbol with parenthesis around the minus signGreek character in Beamer document titleHow to create dashed right arrow over symbol?Currency symbol: Turkish LiraDouble prec as a single symbol?Plus Sign Too Big; How to Call adfbullet?Is there a TeX macro for three-legged pi?How do I get my integral-like symbol to align like the integral?How to selectively substitute a letter with another symbol representing the same letterHow do I generate a less than symbol and vertical bar that are the same height?

          Българска екзархия Съдържание История | Български екзарси | Вижте също | Външни препратки | Литература | Бележки | НавигацияУстав за управлението на българската екзархия. Цариград, 1870Слово на Ловешкия митрополит Иларион при откриването на Българския народен събор в Цариград на 23. II. 1870 г.Българската правда и гръцката кривда. От С. М. (= Софийски Мелетий). Цариград, 1872Предстоятели на Българската екзархияПодмененият ВеликденИнформационна агенция „Фокус“Димитър Ризов. Българите в техните исторически, етнографически и политически граници (Атлас съдържащ 40 карти). Berlin, Königliche Hoflithographie, Hof-Buch- und -Steindruckerei Wilhelm Greve, 1917Report of the International Commission to Inquire into the Causes and Conduct of the Balkan Wars

          Чепеларе Съдържание География | История | Население | Спортни и природни забележителности | Културни и исторически обекти | Религии | Обществени институции | Известни личности | Редовни събития | Галерия | Източници | Литература | Външни препратки | Навигация41°43′23.99″ с. ш. 24°41′09.99″ и. д. / 41.723333° с. ш. 24.686111° и. д.*ЧепелареЧепеларски Linux fest 2002Начало на Зимен сезон 2005/06Национални хайдушки празници „Капитан Петко Войвода“Град ЧепелареЧепеларе – народният ски курортbgrod.orgwww.terranatura.hit.bgСправка за населението на гр. Исперих, общ. Исперих, обл. РазградМузей на родопския карстМузей на спорта и скитеЧепеларебългарскибългарскианглийскитукИстория на градаСки писти в ЧепелареВремето в ЧепелареРадио и телевизия в ЧепелареЧепеларе мами с родопски чар и добри пистиЕвтин туризъм и снежни атракции в ЧепелареМестоположениеИнформация и снимки от музея на родопския карст3D панорами от ЧепелареЧепелареррр