I'm reading Eisenbud's Commutative Algebra with a View Toward Algebraic Geometry, and I am confused by one of his proofs.
The setup is that $R$ is a commutative ring, $U$ is a multiplicatively closed subset, and $\varphi$ is the natural map $\varphi: R \rightarrow R[U^{-1}]$ sending $r$ to $r/1$.
He says that if an ideal $J \subset R$ is of the form $\varphi^{-1}(I)$, where $I \subset R[U^{-1}]$ is an ideal, then "since the elements of $U$ act as units on $R[U^{-1}]/I$, they act as nonzerodivisors on the $R$-submodule $R/J$."
What does he mean by "acting as units on" and "acting as nonzerodivisors on"?