Let $R$ be a commutative unital ring, and let $S \subset R$ be a multiplicative subset. If $P$ is a finitely generated projective $R$-module, then the localization $S^{-1}P$ is a finitely generated projective $S^{-1}R$-module. Consequently, localization induces a [group homomorphism](/page/Group%20Homomorphism)