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.

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.