Algebraic identities play a pivotal role in the study of many mathematical structures although once understood, they are subconsciously regarded as being obvious or even tautological. For instance, polarization identity in convexity results in Hilbert space theory, Sylvester’s determinant identity in the study of determinantal processes, rank identities in the proof of Cochran’s theorem, etc. In this talk, the main goal is to discuss a systematic approach towards developing a theory of rank identities and determinant identities. This makes contact with Cohn’s work on free ideal rings, particularly free associative algebras over a field. By taking a universal approach, we will see how these methods translate to the world of finite von Neumann algebras (specifically II1 factors) where there is a natural notion of center-valued rank which measures the degree of non-degeneracy of an operator, and a notion of determinant known as the Fuglede-Kadison determinant. We will also see some applications to the (non-self-adjoint) algebraic structure of finite von Neumann algebras and to certain operator inequalities.