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Given an associative algebra $A$, there is a correspondence between representations of $A$ and left $A-$ modules. Thus, one can study the representation theory of an associative algebra via its left modules and vice versa.

My question is that, what are the motivations that makes one to use one of the above methods to study $A$ ? What would be the benefits and drawbacks of either cases?

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It's only a matter of emphasis. The representation perspective focuses on the action of the algebra on vectors; whereas, the module perspective focuses on the vectors being acted upon. It all feels a bit tautological.

It's like asking the question about a ceramic artist who throws clay pots: do we concentrate on the potter (the representation) or the pots (the module)? You can't have one without the other.

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