Vector/Endomorphism with this eigenvector/Ring/Dimension/Exercise

Let ${}V$ be a finite-dimensional ${}K$-vector space, and ${}v\in V$, ${}v\neq 0$, a fixed vector. Show that

${}R(v)={\left\{\varphi \in \operatorname {End} _{}{\left(V\right)}\mid v{\text{ is eigenvector of }}\varphi \right\}}\,,$

with the natural addition and multiplication of endomorphisms, is a ring and a linear subspace of ${}\operatorname {End} _{}{\left(V\right)}$. Determine the dimension of this space.

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