Endomorphism Ring

Recall that for a Commutative Monoid $M$, we can form a Semiring $\mathrm{End}(M)$ consisting of all Monoid Homomorphisms $M\to M$ where $f+g=\lambda x.\;f(x)+g(x)$ and $f\cdot g=f\circ g$.

When $M$ is an Abelian Group, this semiring is in fact a Ring, where the additive inverse $-f$ is given by $\lambda x.\;-f(x)$. Moreover, this ring consists of all Group Homomorphisms $M\to M$, as being a Group is a Property of a Monoid.

When $M$ is not Abelian, the addition of two Group Homomorphisms may not be a Group Homomorphism. However, the set of all functions $M\to M$ carries the structure of a (right) Near-Ring.

Properties

• Every ring $R$ embeds into the endomorphism ring of its underlying abelian group. This is essentially a form of the Yoneda Embedding, and in particular Mitchell's Embedding Theorem.