If an Algebra is a vector space over , doesn’t this make it a -module? If so, does it have structure which precludes us from using isomorphism theorems for modules once we have established algebra homomorphisms?

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If an Algebra is a vector space over , doesn’t this make it a -module? If so, does it have structure which precludes us from using isomorphism theorems for modules once we have established algebra homomorphisms?

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Yes!

An algebra is a K-module as well as a ring, so you are free to use isomorphism theorems for modules and rings to get isomorphism theorems for algebras. I said more about this in lecture 3.