This question is about Linear Algebra. I want an example of a ring
which is not a principle ideal domain.

Hi mitran:

Thanks for the interesting question. Always a pleasure to tackle a math question. 

I found the following examples of rings that are *not* principAl ideal
domains (PIDs):

Principal Ideal Domain
URL: http://encyclopedia.laborlawtalk.com/Principal_ideal_domain
Quote: "An example of a non PID is the ring Z[X] of all polynomials
with integer coefficients. It is not principal, since for example the
ideal generated by 2 and X cannot be generated by a single
polynomial."

unique factorization domain 
URL: http://www.answers.com/topic/unique-factorization-domain
Quote: "(Any polynomial ring with more than one variable is an example
of a UFD that is not a principal ideal domain.)"

Search Strategy (on Google):
* example ring "principal ideal domain"

I hope this helps!

websearcher