# Abstract Algebra/Polynomial rings, irreducibility

## Group rings[edit]

**Definition 12.1**:

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

**Failed to parse (unknown function "\middle"): {\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 of ring elements where only finitely many entries are different from zero, together with addition

and multiplication

- .

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[edit]

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