How we can show that a group algebra over infinite cyclic group is a principal ideal domain?

Question

How we can show that a group algebra $\mathbb{F}G$ over infinite cyclic group $G= \langle a\rangle$ is a principal ideal domain?

I tried to calculate the basis of this in the form of ${(a-1)}^n$ type elements....I dont know if this is the approach.

Answer 1

As suggested in Derek's comment, an answer in How to prove the ring of Laurent polynomials over a field is a principal ideal domain? solves the problem. Why? Because the group algebra is isomorphic to the ring of Laurent polynomials.

To see this, consider the function $f:\mathbb{F}G\to \mathbb{F}[x,x^{-1}]$ determined by linearly extending the rule $na\mapsto x^n$ in additve notation, or even more simply $a^n\mapsto x^n$ in multiplicative notation. This is an isomorphism of $\mathbb{F}$-algebras.