Let $R$ be a commutative ring with the unit element. It is shown that an ideal $I$ in $R$ is pure if and only if $\mathrm{Ann}(f)+I=R$ for all $f\in I$. If $J$ is the trace of a projective $R$-module $M$, we prove that $J$ is generated by the “coordinates” of $M$ and $JM=M$. These lead to a few new results and alternative proofs for some known results.