# Abstract Algebra/Polynomial rings, irreducibility

## Group rings

Definition 12.1:

Let ${\displaystyle R}$ be a ring, and let ${\displaystyle G}$ be a group. We can merge them into one object, denoted ${\displaystyle RG}$, called the group ring of ${\displaystyle R}$ and ${\displaystyle G}$ (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 ${\displaystyle G}$ of ring elements where only finitely many entries are different from zero, together with addition

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

and multiplication

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

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.