Are monomorphisms in the category of Artinian rings injective?

Solution 1:

Suppose $f:A\to B$ is a morphism of Artinian rings, and form the kernel pair $A\times_f A=\{(x,y)\in A\times A:f(x)=f(y)\}$ in the category of rings. I claim that $A\times_f A$ is in fact Artinian (so it is also a kernel pair in the category of Artinian rings). Indeed, note that $A\times_f A$ is an $A$-submodule of $A\times A$, and so is Artinian as an $A$-module and therefore also as a module over itself (since the action of $A$ is just via the diagonal subring of $A\times_f A$).

So, if $f$ is a monomorphism in the category of Artinian rings, this in particular implies the two projections $A\times_f A\to A$ must be equal, since they become equal after composing with $f$. It follows that $f$ is injective.