Let $D$ be a noncommutative simple ring with unity that contains no nontrivial zero-divisors, for instance, a Weyl algebra. Let $M(D)$ be the ring of infinite matrices with finitely many nonzero entries over $D$ and let $R = M(D) + DI$, where $I$ is the infinite identity matrix and $DI$ consists of matrices of the form $aI$ with $a \in D$. I need to show that $R$ is a prime ring.
An instructor told me that $R$ is actually a primitive ring. I think that the socle and Jacobson density theorem plays a role here. However, I am far from proving it. Do you know how to prove it?
Note: I tried to show this from the definition of a prime ring, but it became too complex.
Let $R$ act on $V=\oplus_{i=0}^\infty D$ on the left via matrix multiplication. It is not hard to show $V$ is a simple, faithful left $R$ module, hence $R$ is left primitive. Here is a sketch for you to explore:
An $R$ module $S$ is simple iff for every nonzero $x,y\in S$, there exists an $r\in R$ such that $rx=y$.
Use the above to demonstrate $V$ is simple: first note that given any two elements $x,y\in V$, the places they are nonzero (their "supports") is a finite set (of size $n$, say.) You can find an $n\times n$ matrix that moves $x$ to $y$ within the support (this is just basic linear algebra.) Finally, show it can be embedded as an element of $R$ to send $x$ to $y$.
$V$ is clearly faithful (just consider the matrices that are $1$ in exactly one place on the diagonal.)
I leave the details for you to craft.