The implication is false without the assumption that R is Noetherian, because finite matrices don't detect enough information about infinitely generated ideals.

For example, let R be the ring
$$
\bigcup_{n \geq 0} k[[t^{1/n}]]
$$
where $k$ is a field (an indiscrete valuation ring).  Any finite matrix with coefficients in R comes from a subring $k[[t^{1/N}]]$ for some large $N$, and hence can be reduced to Smith normal form within this smaller PID.

However, the ideal $\cup (t^{1/N})$ is not principal.