## FANDOM

36 Pages

Springer GTM 73.

"This book is intended to serve as a basic text for an algebra course at the beginning graduate level. Its writing was begun several years ago when I was unable to find a one-volume text which I considered suitable for such a course."

# Chapter IV: Modules Edit

## Section IV.6: Modules over a Principal Ideal Domain Edit

• Proof of Theorem 6.1: If $c \neq 0$, then the R-module epimorphism $R \mapsto Rc$ of Theorem 1.5(i) is actually an isomorphism. Since R is an integral domain, the kernel of this map is zero, so the map is injective. Consequently, any ideal I of a PID R is isomorphic, as an R-module, to R. (The potentially frightening implications in the finite case are dismissed by recalling that finite integral domains are already fields.)