Prove the ring equivalent of Cayley's theorem [duplicate]

Show that for every ring $(R,+,\cdot)$, there is an abelian group, $(A,+)$, such that $R$ is isomorphic to a subring of $(\operatorname{End}(A),+,\circ)$.

$(\operatorname{End}(A),+,\circ)$ is the set of homomorphisms of $A$ that form a ring under function addition and composition.

I am thinking to let $\operatorname{End}(A)$ be the group such that $A$ is the abelian group $(R,+)$ and create a ring homomorphism from $(R,+,\cdot)$ into $\operatorname{End}((R,+))$.

Thoughts?

Solution 1:

You are correct, (madame or) sir.

This is essentially the ring-theoretic analogue of Cayley's Theorem for groups.

Also, this issue (as a question) came up a while back on Math Overflow.

Added: I had missed that the explicit definition of the map was not contained in the OP's question. A natural ring embedding from $R$ to $\operatorname{End}(R,+)$ is

$r \mapsto \bullet r: (x \in R \mapsto xr)$.

[Or possibly $r \mapsto r \bullet: (x \in R \mapsto rx)$, depending upon your conventions on composition.]

Solution 2:

I think instead of using $(R,+)$ as the candidate abelian group, you should use $$ A = \oplus_{a\in R}A_a $$ where $A_a$ is the cyclic group generated by the element $a$.

Let $f \in End(A)$. Then we can think of $f$ as $$ \oplus_{a\in R}f_a $$ where $f_a \in End(A_a)$.

Finally, we can define a map $\phi: R \rightarrow End(A)$ such that for $b \in R$ $$ \phi(b) = \oplus_{a\in R}f_a $$ where $f_a$ is the trivial map if $a \neq b$; and $f_a$ is the identity map if $a = b$.