Question

Give an example of a non-PID over which every finitely generated module is a direct sum of cyclic modules.

We do this by finding a ring R that is not an integral domain. Then use the fundamental theorem of finite abelian groups.

Answer 1

ANSWER:

Solution It is known that finitely generated modules over a PID . There Non-PID can be writt as form pta po rl(d) --- Rkan) generalizing this, we say finitely generated module 'M' over orbitary Commutati Can written in this form of MŠ Bf & R) 1,0 --- R 120 where Do is a ideals g R. Notation - R - RO... BR means f- fold direct sum "R' with itself. - there even for generalization PID this longer fome. This problem domain Cany many Other types g rings) there are projective modules that are not free. the fundamental theorem are use finite a belian groups

funcdamental paramp theoram modules domain R. let nzo M tours be torsion g sub module Rinitely generated rank M and - MERO-BRO 1 6 Mars I N factors ideal 2 and R M ERIPCX ---X RIALS some s>o and powers of prime ideals pi the ideals per unique nique and and id ideal I unique up multiplication by principal ideal. NOTE : to with respect to additior NOTE : Everly ring abelian group with respect to ad Operation. In commutative dring the invertible Elements (or) units, from an abelian multiplicative group. In particular real numbers and abelian group under addition, the Mon zero real Numbeos are abelian group undeor multiple cation.