Multiplicative group of rational $\mathbb Q^*\cong \mathbb Z_2 \times \bigoplus_{i=1}^{\infty} \mathbb Z$.
My question:
In general, how to characterize the multiplicative group $K^∗$ of the fraction field $K$ of a UFD $R$ ?
Can we use the same method which was used in $\mathbb Q^*$?
And is there any reference about this?
Thank you for your time and help!
Your general statement is totally false (note that the answer you linked stated that description for a number field, not for the field of fractions of an arbitrary UFD). For one thing, the description you gave would say that $K^*$ is always countable, which is certainly wrong. It's not even correct if you remove the restriction that the free part has countable rank: for instance, $\mathbb{C}$ is a UFD and is its own fraction field, and its multiplicative group has no nontrivial cyclic group as a direct summand (since it is divisible).
In general, if $R$ is a UFD, then the multiplicative group of the field of fractions of $R$ is isomorphic to $R^\times\times \bigoplus_I \mathbb{Z}$ where $I$ is the set of irreducible elements of $R$ (modulo units). The proof is exactly the same as in the case of $\mathbb{Q}$. Note though that the group $R^\times$ of units in $R$ can be much more complicated than just a cyclic group and $I$ could have any cardinality.