Retract

Remarks

Up to isomorphism, an algebra $\mathbf{C}$ is a retract of $\mathbf{A}$ when there are homomorphisms $s \colon \mathbf{C} \to \mathbf{A}$ and $r \colon \mathbf{A} \to \mathbf{C}$ with $r \circ s = \id_C$. Then $s$ is injective and $s(\mathbf{C})$ is a retract of $\mathbf{A}$ in the sense above.

Example: in the group $\Z \times \Z$, the subgroup $\Z \times \set{0}$ is a retract. The map $r(m, n) = (m, 0)$ is a homomorphism onto $\Z \times \set{0}$ and fixes each $(m, 0)$.

Non-example: the subgroup $2\Z$ of $\Z$ is not a retract. A homomorphism $r \colon \Z \to 2\Z$ is determined by $x = r(1) \in 2\Z$, and then $r(2) = 2x$. The condition $r(2) = 2$ forces $x = 1 \notin 2\Z$.