Fraleigh (A First Course in Abstract Algebra) defines principal ideals in section 27 on page 250. His definition is as follows:
If R is a commutative ring with unity and , the ideal of all multiples of a is the principal ideal generated by a and is denoted <a>.
An ideal N of R is a principal ideal if N = <a> for some
If we take r = a in (1) then we have
If we take r = a and the we have using (1) again that
Continuing this, then we have , .... all belonging to N along with the other elements where
Is the above analysis correct regarding the nature of principal ideals?
Would really appreciate this issue being clarified.