Krull Schmidt, An Introduction

Introduction

The Jordan Holder theorem states that a group, with a decomposition series, presents a unique list of factor groups along the way. If you start with a different kernel, and build a different decomposition series, you get the same factor groups, possibly in a different order. This is similar to unique factorization of integers into primes.

Jordan Holder was generalized to modules without too much fuss. The quotient modules that arise from the decomposition series of M are uniquely a function of M. A different decomposition series will produce the same factors, possibly in a different order.

The Krull Schmidt theorem is similar. Write G as a direct product of smaller groups, and if the component groups are indecomposable, i.e. not a direct product of still smaller groups, then the decomposition is unique. There is only one way to write G as the finite direct product of component groups. For instance, Z₃Z₅Z₂₀ is the direct product of Z₃, Z₅, and Z₂₀, and there is no other way to write this as a direct product of indecomposable groups.

Like Jordan Holder, Krull Schmidt generalizes to modules. A module M is the direct product of submodules in a unique way. As you move through the proof, take a moment to verify each step, as it is applied to modules. In many cases the algebra is easier, because the group suddenly becomes abelian. You only need turn group homomorphisms into module homomorphisms, and you're home free.