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He is well known for providing a simple proof of the Quillen–Suslin theorem, a result in commutative algebra, first conjectured by Jean-Pierre Serre in 1955, and then proved by Daniel Quillen and Andrei Suslin in 1976.
Going up and going down, terms in commutative algebra which refer to certain properties of chains of prime ideals in integral extensions
Hilbert's basis theorem, in commutative algebra, stating every ideal in the ring of multivariate polynomials over a Noetherian ring is finitely generated
Hilbert's syzygy theorem, a result of commutative algebra in connection with the syzygy problem of invariant theory
Regular sequence, which is an important topic in commutative algebra.