Injective algebra

Remarks

When $\mathcal{K}$ is a variety, the embeddings are exactly the monomorphisms of the category of algebras of $\mathcal{K}$. So in a variety, $A$ is injective in this sense exactly when it is an injective object of that category.

Example: $\Q$ is injective in the variety of abelian groups. An abelian group is injective there exactly when it is divisible, and $\Q$ is divisible since $x = n \cdot (x/n)$ for every $x \in \Q$ and $n \ge 1$. In the variety of modules over a ring, the injective algebras are the injective modules.

Non-example: $\Z$ is not injective in the variety of abelian groups. Let $g \colon \Z \to \Q$ be the inclusion and $f = \id_{\Z}$. A homomorphism $h \colon \Q \to \Z$ with $h(1) = 1$ would give $2 h(1/2) = 1$ in $\Z$, which is impossible.

In many varieties, such as groups, only the trivial algebra is injective.