Abstract Algebra/Polynomial rings, irreducibility

Group ringsEdit

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.

ExercisesEdit

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