The Nature of Principal Ideals

Fraleigh (A First Course in Abstract Algebra) defines principal ideals in section 27 on page 250. His definition is as follows:

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"27.21 Definition

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

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Consider ...........................(1)

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.

Peter

Re: The Nature of Principal Ideals

Yes. An ideal is also closed under addition (in fact, it is a subgroup of ), so you also get elements such as with . However, you can see that this is the same as , where . Since multiplication by an element in and adding elements in generates the whole ideal, this shows (by example, not proof) that

Re: The Nature of Principal Ideals

Thanks Gusbob ... That post was really helpful in characterizing the other elements you mentioned!!!

Appreciate your help

Peter