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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 $(f\cdot g)(x)=f(g(x))$
  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?

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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.

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