Let $R$ be any commutative ring with $1$, and let $M$ be any finitely generated torsion-free $R$-module. Can we embed $M$ into a free $R$-module of finite rank?
As we know that it can be embedded if $R$ is a domain. Since we can then take the tensor product of $M$ with $K$ (the quotient field of $R$) and find suitable basis so that we can embed. But I don’t know whether it is true for any commutative ring or not?