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 $\displaystyle a \in R $ , the ideal $\displaystyle \{ ra | r \in R \} $ 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 $\displaystyle a \in R $

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Consider $\displaystyle N =\{ ra | r \in R \} $ ...........................(1)

If we take r = a in (1) then we have $\displaystyle ra = aa = a^2 \in N $

If we take r = a and $\displaystyle a^2 \in N $ the we have using (1) again that $\displaystyle ra = a^2 a = a^3 \in N$

Continuing this, then we have $\displaystyle a, a^2, a^3, a^4, a^5 $ , .... all belonging to N along with the other elements where $\displaystyle r \ne a $

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 $\displaystyle (R,+)$), so you also get elements such as $\displaystyle a^2+ a^7 +ra+sa $ with $\displaystyle r,s\in R$. However, you can see that this is the same as $\displaystyle (a+a^6+r+s)a$, where $\displaystyle (a+a^6+r+s)\in R$. Since multiplication by an element in $\displaystyle R$ and adding elements in $\displaystyle N$ generates the whole ideal, this shows (by example, not proof) that $\displaystyle \langle a \rangle = \{ra|r\in R\}$

Re: The Nature of Principal Ideals

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

Appreciate your help

Peter