Must a local homomorphism from a Noetherian local ring to an artinian local ring factor through a power of its maximal ideal

Question

Let $f : B\rightarrow A$ be a local homomorphism of Noetherian local rings where $A$ is moreover Artinian. Must $f$ factor through $B/m_B^n$ for some $n$?

Answer 1

Yes. Note that $m_A^n=0$ for some $n$, since $A$ is Artinian. Also $f(m_B)\subseteq m_A$ (local homomorphism) so $f(m_B^n)\subseteq m_A^n=0$.