My professor defined an algebra over a field as a vector space over that field that is also a ring. He did not state however with regard to what operations is the algebra a ring so can I assume the addition in the algebra is the usual addition in the sense of the algebra as a vector space over the field?
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3You guessed absolutely rigth (and it should have been clarified). Beyond that, the multiplication in the ring is compatible with the multiplication of vectors by a scalar – Hagen von Eitzen Mar 03 '21 at 16:58
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Also see my answer to this question which might help you along the way. – Bumblebee Mar 03 '21 at 16:59