Let $D$ be a division ring, and $V$ a vector space over $D$, such that $\dim_DV=n$ where $2\le n<+\infty$.
Then I must prove that $R:=\operatorname{End}(V)$ is a prime ring (a ring $R$ is prime if $aRb=0\;\;\Rightarrow\;\;a=0\;\vee\;b=0$) but not an integral domain.
I'm pretty sure about the correctness of my argument (which won't be written). My doubt is about the following preliminary consideration, on which the whole argument is based.
Every couple of $n-$dimensional $D$ vector space are isomorphic, hence every $n-$dimensional $D$ vector space is isomorphic to $D^n$.
Thus $R:=\operatorname{End}(V)\simeq\operatorname{End}(D^n)\simeq M_n(D)$ (this last one is the ring of $n\times n$ matrices over $D$).
Thus I simply worked with $M_n(D)$ to prove the initial claim. Is this correct?