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

Question

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

Angina Seng · Answer

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

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

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