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?

  • $\begingroup$ Perhaps mathoverflow.net/questions/348736/… answers your question. $\endgroup$
    – efs
    Feb 21, 2020 at 15:18
  • $\begingroup$ @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 $\endgroup$
    – Yemon Choi
    Feb 21, 2020 at 15:23

1 Answer 1


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!


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