# Left module which cannot be made into a bimodule?

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?

• Perhaps mathoverflow.net/questions/348736/… answers your question. – EFinat-S Feb 21 at 15:18
• @EFinat-S No, because that question only asks for the existence of some right $A$-module structure that is incompatible with the given left $A$-module structure. This question is asking for something much stronger – Yemon Choi Feb 21 at 15:23

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!