Let $S=k[[t^3,t^5,t^7]]$ be a formal power series over field $k$.I wanna know why $\dim_k \operatorname{Soc}(S/t^3S)=2?$.($\dim_k$ means dimension as $k$-vector space.)
background: $\operatorname{Soc}(M)=(0:_M m)=\lbrace x\in M ; xm=0 \rbrace$ where $(R,m)$ is a local ring and $M$ is an $R$-module.