Let $R$ be an (associative, unital) ring. By an $R$-module we mean a left $R$-module. We call an $R$-module torsionless if it can be embedded into a direct product of the regular $R$-module $R$.
I am looking for a Noetherian ring $R$ that is an injective $R$-module over itself which has the property that not every $R$-module is torsionless. Does such a ring exist? What would be an example? Where should I look for an example?
A ring that is Noetherian on a side and self-injective on a side is quasi-Frobenius. The Faith-Walker theorem says that every injective module of a QF ring embeds in a free module. Since every module embeds in its injective envelope, we get that every module over a quasi-Frobenius ring is torsionless.
So no, there is no such example.