Let $A$ be a unital algebra, defined over the complex numbers. Any bimodule $M$ over $A$ must, by definition, be a left, and right, module satisfing
$$
a.(m.b) = (a.m).b, ~~~~~~~ \textrm{ for all } a,b \in A, ~ m \in M
$$
What is a "natural" or "well-motivated" example of a an object which is both a left and right module, but does not satisfy the above condition?



Left-right non-bimodule examples
Suppose $A$ is a non-commutative Hopf algebra. Then you can use $M=A$ with the left adjoint and right regular actions:
$$
a\cdot m = \sum\_{(a)}a\_{(1)}mS(a\_{(2)}), \ m\cdot b = mb .
$$
In particular, you can use the group algebra of a a group $G$ so that
$$
g\cdot m = gmg^{-1}, \ m\cdot h = mh, \ g,h,m\in G.
$$


