# Rank of completion of a module

Let $(R,m)$ be a Noetherian local ring and $M$ a finitely generated $R$-module. Let $\hat R$ denote the $m$-adic completion of $R$ and $\hat M\cong M\otimes {\hat R}.$

Is there any relation between rank$_R{M}$ and rank$_{\hat R}{\hat M}.$

• What do you mean by rank? – neilme Feb 12 '18 at 18:33
• They are equal. – user26857 Feb 12 '18 at 20:50
• @user26857 Could you please explain it. It is not clear to me. – Cusp Feb 13 '18 at 4:18
• Why do people always ask what the rank is? Over every commutative ring with unity, you can always define it as the maximal size of a linear independent subset. If the ring gets more convenient, one has other definitions. But at the very least, one has this basic definition. – MooS Feb 13 '18 at 8:35
• By flatness the rank can only increase. By faithfully flatness, the rank cannot increase. – MooS Feb 13 '18 at 9:05