A conceptual question:
Let's say there's a linear matrix representation of a particular algebra. I'm wondering just how much this matrix representation can tell us about the 'outstanding' elements of the algebra (nilpotents, idempotents, zero divisors etc). It seems clear that the number of these elements would be the same in the representation as in the algebra. But would the representation preserve the conditions of existence of the elements? I.e., would the conditions of existence for idempotent and nilpotent elements (and/or zero divisors) be the same in the isomorphic matrix representation and in the algebra?
In a more abstract, overarching sense: how far exactly do matrix representations go in preserving algebraic structure? How should one conceive of matrix representations to make the answers to these questions manifest?