Much of my recent work on general operator algebras has been focused on finding ways that finite dimensional algebra, i.e., ring theory, can illuminate avenues of investigation in the infinite dimensional setting. The goal is to understand how pure algebra interacts with metric constraints. Even in the setting of finite dimensional algebras, there is still a lot that is mysterious. This talk will illustrate this endevor by describing how some observations of Hochschild from 1947 have inspired some more recent work in the structure and representation theory of certain non-self-adjoint operator algebras.