O'Reilly logo

Basic Abstract Algebra by S. K. Jain, S. R. Nagpaul, P. B. Bhattacharya

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

CHAPTER 21

Finitely generated modules over a PID

Our object in this chapter is to prove the fundamental structure theorem for finitely generated modules over a principal ideal domain. This theorem provides a decomposition of such a module into a direct sum of cyclic modules. In particular, it gives an independent proof of the fundamental theorem of finitely generated abelian groups proved earlier in Chapter 8. Applications of this theorem to linear algebra are given in Sections 4 and 5.

1 Decomposition theorem

1.1 Theorem. Let R be a principal ideal domain, and let M be any finitely generated R-module. Then

images

a direct sum of cyclic modules, ...

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required