Editing Principal ideal domain (section)

==Modules==
{{main|Structure theorem for finitely generated modules over a principal ideal domain}}
The key result is the structure theorem:  If ''R'' is a principal ideal domain, and ''M'' is a finitely
generated ''R''-module, then <math>M</math> is a direct sum of cyclic modules, i.e., modules with one generator.  The cyclic modules are isomorphic to <math>R/xR</math> for some <math>x\in R</math><ref>See also Ribenboim (2001), [https://books.google.com/books?id=u5443xdaNZcC&pg=PA113 p. 113], proof of lemma 2.</ref> (notice that <math>x</math> may be equal to <math>0</math>, in which case <math>R/xR</math> is <math>R</math>).

If ''M'' is a [[free module]] over a principal ideal domain ''R'', then every submodule of ''M'' is again free.<ref>[https://people.math.sc.edu/mcnulty/algebra/grad/pidfree.pdf Lecture 1. Submodules of Free Modules over a PID] math.sc.edu Retrieved 31 March 2023</ref><!-- need a better reference; for example, from a textbook --> This does not hold for modules over arbitrary rings, as the example <math>(2,X)  \subseteq \mathbb{Z}[X]</math>   of  modules over <math>\mathbb{Z}[X]</math> shows.