Let $A$ be a noncommutative unital algebra, defined over $\mathbb{C}$ say. What is an example of a left $A$-module $M$ that does not admit a right $A$-module structure giving $M$ the structure of a bimodule?



Left module which cannot be made into a bimodule?
Such examples are a plenty. You are asking about non-existence of an algebra map $A\rightarrow End\_AM$. Take $A$ simple, at most countably dimensional, and a simple module ${}\_{A}M$. Then $End\_AM={\mathbb C}$. Bingo!


