What algebraic structure does the set of endomorphisms of a ring have?












2












$begingroup$


Let $R$ be a ring, and let $End(R)$ be the set of ring endomorphisms of $R$, i.e. the set of all ring homomorphisms form $R$ to $R$. Then we can define three binary operations on $End(R)$:





  1. $+$, defined by $(f+g)(x)=f(x)+g(x)$


  2. $cdot$, defined by $(fcdot g)(x)=f(g(x))$


  3. $*$, defiend by $(f*g)(x)=f(x)g(x)$


Now $(End(R),+,cdot)$ is a ring, being a subring of the endomorphism ring of the abelian group $(R,+)$. But my question is, what is the algebraic structure of $(End(R),+,cdot,*)$? Does this beast with three binary operations have a name?



Also, what algebraic structure does $(End(R),+,*)$ have? Is that also a ring?






abstract-algebra ring-theory modules terminology ring-homomorphism







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    en.wikipedia.org/wiki/Composition_ring
    $endgroup$
    – Chris Culter
    2 hours ago






  • 1




    $begingroup$
    @ChrisCulter Thanks, I guess that's the notion I was groping towards. I just posted a follow-up question: math.stackexchange.com/q/3075029/71829
    $endgroup$
    – Keshav Srinivasan
    1 hour ago
















2












$begingroup$


Let $R$ be a ring, and let $End(R)$ be the set of ring endomorphisms of $R$, i.e. the set of all ring homomorphisms form $R$ to $R$. Then we can define three binary operations on $End(R)$:





  1. $+$, defined by $(f+g)(x)=f(x)+g(x)$


  2. $cdot$, defined by $(fcdot g)(x)=f(g(x))$


  3. $*$, defiend by $(f*g)(x)=f(x)g(x)$


Now $(End(R),+,cdot)$ is a ring, being a subring of the endomorphism ring of the abelian group $(R,+)$. But my question is, what is the algebraic structure of $(End(R),+,cdot,*)$? Does this beast with three binary operations have a name?



Also, what algebraic structure does $(End(R),+,*)$ have? Is that also a ring?






abstract-algebra ring-theory modules terminology ring-homomorphism







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    en.wikipedia.org/wiki/Composition_ring
    $endgroup$
    – Chris Culter
    2 hours ago






  • 1




    $begingroup$
    @ChrisCulter Thanks, I guess that's the notion I was groping towards. I just posted a follow-up question: math.stackexchange.com/q/3075029/71829
    $endgroup$
    – Keshav Srinivasan
    1 hour ago














2












2








2





$begingroup$


Let $R$ be a ring, and let $End(R)$ be the set of ring endomorphisms of $R$, i.e. the set of all ring homomorphisms form $R$ to $R$. Then we can define three binary operations on $End(R)$:





  1. $+$, defined by $(f+g)(x)=f(x)+g(x)$


  2. $cdot$, defined by $(fcdot g)(x)=f(g(x))$


  3. $*$, defiend by $(f*g)(x)=f(x)g(x)$


Now $(End(R),+,cdot)$ is a ring, being a subring of the endomorphism ring of the abelian group $(R,+)$. But my question is, what is the algebraic structure of $(End(R),+,cdot,*)$? Does this beast with three binary operations have a name?



Also, what algebraic structure does $(End(R),+,*)$ have? Is that also a ring?






abstract-algebra ring-theory modules terminology ring-homomorphism







share|cite|improve this question









$endgroup$




Let $R$ be a ring, and let $End(R)$ be the set of ring endomorphisms of $R$, i.e. the set of all ring homomorphisms form $R$ to $R$. Then we can define three binary operations on $End(R)$:





  1. $+$, defined by $(f+g)(x)=f(x)+g(x)$


  2. $cdot$, defined by $(fcdot g)(x)=f(g(x))$


  3. $*$, defiend by $(f*g)(x)=f(x)g(x)$


Now $(End(R),+,cdot)$ is a ring, being a subring of the endomorphism ring of the abelian group $(R,+)$. But my question is, what is the algebraic structure of $(End(R),+,cdot,*)$? Does this beast with three binary operations have a name?



Also, what algebraic structure does $(End(R),+,*)$ have? Is that also a ring?






abstract-algebra ring-theory modules terminology ring-homomorphism




abstract-algebra ring-theory modules terminology ring-homomorphism






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked 2 hours ago









Keshav SrinivasanKeshav Srinivasan

2,11811441




2,11811441








  • 1




    $begingroup$
    en.wikipedia.org/wiki/Composition_ring
    $endgroup$
    – Chris Culter
    2 hours ago






  • 1




    $begingroup$
    @ChrisCulter Thanks, I guess that's the notion I was groping towards. I just posted a follow-up question: math.stackexchange.com/q/3075029/71829
    $endgroup$
    – Keshav Srinivasan
    1 hour ago














  • 1




    $begingroup$
    en.wikipedia.org/wiki/Composition_ring
    $endgroup$
    – Chris Culter
    2 hours ago






  • 1




    $begingroup$
    @ChrisCulter Thanks, I guess that's the notion I was groping towards. I just posted a follow-up question: math.stackexchange.com/q/3075029/71829
    $endgroup$
    – Keshav Srinivasan
    1 hour ago








1




1




$begingroup$
en.wikipedia.org/wiki/Composition_ring
$endgroup$
– Chris Culter
2 hours ago




$begingroup$
en.wikipedia.org/wiki/Composition_ring
$endgroup$
– Chris Culter
2 hours ago




1




1




$begingroup$
@ChrisCulter Thanks, I guess that's the notion I was groping towards. I just posted a follow-up question: math.stackexchange.com/q/3075029/71829
$endgroup$
– Keshav Srinivasan
1 hour ago




$begingroup$
@ChrisCulter Thanks, I guess that's the notion I was groping towards. I just posted a follow-up question: math.stackexchange.com/q/3075029/71829
$endgroup$
– Keshav Srinivasan
1 hour ago










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$begingroup$

Of your three operations, only $cdot$ is actually a valid operation. In general, if $f$ and $g$ are ring endomorphisms, then your $f+g$ and $f*g$ are not ring endomorphisms: $f+g$ will typically not preserve multiplication and $f*g$ typically will not preserve addition (or multiplication, if $R$ is not commutative).



The natural structure that $operatorname{End}(R)$ has under the composition operation $cdot$ is a monoid: composition is associative and has an identity element (the identity map). This is not special to rings but is true of the set of endomorphisms of pretty much any kind of mathematical object.






share|cite|improve this answer









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    $begingroup$

    Of your three operations, only $cdot$ is actually a valid operation. In general, if $f$ and $g$ are ring endomorphisms, then your $f+g$ and $f*g$ are not ring endomorphisms: $f+g$ will typically not preserve multiplication and $f*g$ typically will not preserve addition (or multiplication, if $R$ is not commutative).



    The natural structure that $operatorname{End}(R)$ has under the composition operation $cdot$ is a monoid: composition is associative and has an identity element (the identity map). This is not special to rings but is true of the set of endomorphisms of pretty much any kind of mathematical object.






    share|cite|improve this answer









    $endgroup$


















      6












      $begingroup$

      Of your three operations, only $cdot$ is actually a valid operation. In general, if $f$ and $g$ are ring endomorphisms, then your $f+g$ and $f*g$ are not ring endomorphisms: $f+g$ will typically not preserve multiplication and $f*g$ typically will not preserve addition (or multiplication, if $R$ is not commutative).



      The natural structure that $operatorname{End}(R)$ has under the composition operation $cdot$ is a monoid: composition is associative and has an identity element (the identity map). This is not special to rings but is true of the set of endomorphisms of pretty much any kind of mathematical object.






      share|cite|improve this answer









      $endgroup$
















        6












        6








        6





        $begingroup$

        Of your three operations, only $cdot$ is actually a valid operation. In general, if $f$ and $g$ are ring endomorphisms, then your $f+g$ and $f*g$ are not ring endomorphisms: $f+g$ will typically not preserve multiplication and $f*g$ typically will not preserve addition (or multiplication, if $R$ is not commutative).



        The natural structure that $operatorname{End}(R)$ has under the composition operation $cdot$ is a monoid: composition is associative and has an identity element (the identity map). This is not special to rings but is true of the set of endomorphisms of pretty much any kind of mathematical object.






        share|cite|improve this answer









        $endgroup$



        Of your three operations, only $cdot$ is actually a valid operation. In general, if $f$ and $g$ are ring endomorphisms, then your $f+g$ and $f*g$ are not ring endomorphisms: $f+g$ will typically not preserve multiplication and $f*g$ typically will not preserve addition (or multiplication, if $R$ is not commutative).



        The natural structure that $operatorname{End}(R)$ has under the composition operation $cdot$ is a monoid: composition is associative and has an identity element (the identity map). This is not special to rings but is true of the set of endomorphisms of pretty much any kind of mathematical object.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 2 hours ago









        Eric WofseyEric Wofsey

        181k12208336




        181k12208336






























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