Difficult# Endomorphism Ring of an Abelian Group

ABSALG-GLZX0E

An *endomorphism* of a group $G$ is a group homomorphism $h : G \rightarrow G$.

If $G$ is an **abelian** group, then we can define a ring structure on the endomorphisms of $G$, and we denote this ring by $\text{End}(G)$.

Addition in the ring is defined by

$(h_1+h_2)(g)=h_1(g)+h_2(g)$, $\quad h_1,h_2\in \text{End}(G)$, $\quad g\in G$

...that is, by the pointwise addition of endomorphisms.

Multiplication in the ring is defined by

$(h_1\circ h_2)(g)=h_1(h_2(g))$, $\quad h_1,h_2\in \text{End}(G)$, $\quad g\in G$

...that is, by the composition of endomorphisms.

Which of the following are true statements about $\text{End}(\mathbb{Z} \times \mathbb{Z})$?