Let D be a division ring and V a vector space over D of dimension at least 2. I have to show that End(V) is a prime ring and is not an integral domain.
($R$ is a prime ring if, for every $a,b \in R$, $aRb=0$ implies $a=0$ or $b=0$)
I have no idea how do it! Thanks for your help!
To show End (V) is not an integral domain, just take $f,g\in End (V)\setminus\{0\}$ such that $ker (f)=im (g)\neq\{0\} $.
Applying the definition of primeness you gave is not too bad here. Show that if $a$ and $b$ are nonzero linear transformations, then you can design a linear transformation $c$ such that $acb\neq 0$. Just strategically select sommething that $a$ does not map to zero, and something nonzero in the image of $b$, and create a linear transformation that makes the composition nonzero.
It's very easy, for example, to create linear transformations $T$ such that $T^2=0$. This would be enough to show that the ring is not a domain.