# Abstract Algebra/Polynomial rings, irreducibility

## Group rings

Definition 12.1:

Let $R$ be a ring, and let $G$ be a group. We can merge them into one object, denoted $RG$ , called the group ring of $R$ and $G$ (which is a ring), by taking the fundamental set

$\displaystyle \left\{ (r_g)_{g \in G} \middle| r_g \neq 0 \text{ only for finitely many } g \in G \right\}$ ,

that is, tuples over $G$ of ring elements where only finitely many entries are different from zero, together with addition

$(r_{g})_{g\in G}+(s_{g})_{g\in G}:=(r_{g}+s_{g})_{g\in G}$ and multiplication

$(r_{g})_{g\in G}\cdot (s_{g})_{g\in G}:=\left(\sum _{fh=g}r_{f}s_{h}\right)_{g\in G}$ .

It is a straightforward exercise to show that this is, in fact, a ring. The same construction can be carried out with monoids instead of rings; it is completely the same with all definitions carrying over. In this case, we speak of the monoid ring.

### Exercises

• Exercise 12.1.1: Prove that a group ring and a monoid ring is, in general, a ring.