Goldie studied on a scholarship at St. John's College, Cambridge, where he graduated in 1941, the Tripos examinations for Best. During World War II he worked on the use of CP Snow back in ballistics, but turned at the same time under the influence of Philip Hall of algebra. From 1946 he was Assistant Lecturer at the University of Nottingham since 1948 Lecturer at the University of Newcastle and from 1963 until his retirement in 1986 Professor of Pure Mathematics at the University of Leeds.
From the 1950s he busied himself with ring theory, first with FF Bonsall in the functional analysis, then with non-commutative rings in algebra. Here he made a significant contribution theorem named after him.