The Göllnitz-Gordon-Andrews identities generalize the partition identities discovered independently by H. Göllnitz and B. Gordon. In this article, we present a commutative algebra proof of the Göllnitz-Gordon-Andrews identities. More generally, we establish a family of identities, the special cases of which are the Göllnitz-Gordon-Andrews identities. In the proof, we relate the generating functions associated with these identities to the Hilbert-Poincaré series of suitably constructed graded algebras.