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Abstract Algebra

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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 $\text{End}(\mathbb{Z} \times \mathbb{Z})$?

A

It is isomorphic to $\mathbb{Z} \times \mathbb{Z}$.

B

It is isomorphic to $M_2(\mathbb{Z})$.

C

It is commutative.

D

It is an integral domain.

E

It has infinitely many distinct ideals.