In this post I'll describe the structure theorem over PID's which generalizes the following results: Finite dimensional vector fields over $latex {k}&fg=000000$ are all of the form $latex {k^{\oplus n}}&fg=000000$, The classification theorem for finitely generated abelian groups, The Frobenius normal form of a matrix, The Jordan decomposition of a matrix. 1. Some ring theory… Continue reading The Structure Theorem over PID’s