# commutative algebra – Is \$widehat{IM}=widehat I widehat M? \$ for any finitely generated module \$M\$ , where \$widehat {(-)}\$ denotes completion w.r.t. maximal ideal?

Let $$(R,mathfrak m)$$ be a Noetherian local ring, and let $$(widehat R, widehat{mathfrak m})$$ be its $$mathfrak m$$-afic completion. Let $$M$$ be a finitely generated $$R$$-module.

Then, is it true that $$widehat{IM}=widehat I widehat M?$$

I know this is true if $$I=mathfrak m$$ as explained here If \$M\$ be a finitely generated module over a Noetherian ring \$A,\$ then \$widehat{aM}=hat{a} hat{M}.\$ , but I’m not sure what happens for general $$I$$. Please help.