What should I call an "injective" algebra?

Question

Given rings $A,B$, we say that $B$ is an $A$-algebra if there is a ring homomorphism $f:A\rightarrow B$. This homomorphism give the structure of the algebra. Various properties of algebras can either be said of $B$ or of $f$ (such as being flat, finite, finite type, integral, etc). Is there a standard term to describe $B$ when $f$ is injective? For example, "$B$ is a(n) [adjective] $A$-algebra".

Answer 1

I have seen this called a faithful $A$-algebra, though I am not sure how universal this terminology is. This comes from the much more common term "faithful $A$-module", which is an $A$-module $M$ such that for each nonzero $a\in A$ there exists $m\in M$ such that $am\neq 0$. If $B$ is an $A$-algebra, then it is easy to see that it is faithful as an $A$-module iff the homomorphism $f:A\to B$ is injective.