The Stacks project

Example 35.4.7. For a ring $R$ and $f_1, \ldots , f_ n \in R$ generating the unit ideal, the morphism $R \to R_{f_1} \oplus \ldots \oplus R_{f_ n}$ is universally injective. Although this is immediate from Lemma 35.4.8, it is instructive to check it directly: we immediately reduce to the case where $R$ is local, in which case some $f_ i$ must be a unit and so the map $R \to R_{f_ i}$ is an isomorphism.