If $M\cong N \oplus k^{\oplus a}$, where $a=\dim_k \dfrac{soc(M)}{soc(M)\cap \mathfrak m M }$, then $k$ is not a direct summand of $N$?

Question

Let $M$ be a finitely generated module over a commutative Noetherian local ring $(R,\mathfrak m, k)$ such that $M\cong N \oplus k^{\oplus a}$, where $a=\dim_k \dfrac{soc(M)}{soc(M)\cap \mathfrak m M }$.

Then, is it true that $soc(N)\subseteq \mathfrak m N$? Equivalently, is it true that $k$ is not a direct summand of $N$?

Here, let me recall that for an $R$-module $M$, we set $soc(M):=(0:_M \mathfrak m) $.

Answer 1

This is straight forward.

$soc(M)=soc(N)\oplus k^a$ and $\mathfrak{m}M=\mathfrak{m}N$. So, $soc(M)\cap \mathfrak{m} M=soc(N)\cap \mathfrak{m}N$. Thus you get $soc(N)/soc(N)\cap\mathfrak{m} N=0$, which is what you want.