“Why should we want to know about representations over rings that are not fields of characteristic zero? It is because they arise in many parts of mathematics. Group representations appear any time we have a group of symmetries where there is some linear structure present, over some commutative ring. That ring need not be a field of characteristic zero.
Here are some examples.
- In the theory of error-correcting codes many important codes have a non-trivial symmetry group and are vector spaces over a finite field, thereby providing a representation of the group over that field.”