The Stacks project

Remark 33.38.9. Let $A$ be a reduced ring. Let $I, J$ be ideals of $A$ such that $V(I) \cup V(J) = \mathop{\mathrm{Spec}}(A)$. Set $B = A/J$. Then $I \to IB$ is an isomorphism of $A$-modules. Namely, we have $IB = I + J/J = I/(I \cap J)$ and $I \cap J$ is zero because $A$ is reduced and $\mathop{\mathrm{Spec}}(A) = V(I) \cup V(J) = V(I \cap J)$. Thus for any projective $A$-module $P$ we also have $IP = I(P/JP)$.