  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 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.
