# The Nature of Principal Ideals

#### Peter

##### Well-known member
MHB Site Helper
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 [TEX] a \in R [/TEX] , the ideal [TEX] \{ ra | r \in R \} [/TEX] 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 [TEX] a \in R [/TEX]

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Consider [TEX] N =\{ ra | r \in R \} [/TEX] ...........................(1)

If we take r = a in (1) then we have [TEX] ra = aa = a^2 \in N [/TEX]

If we take r = a and [TEX] a^2 \in N [/TEX] the we have using (1) again that [TEX] ra = a^2 a = a^3 \in N[/TEX]

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

Is the above analysis correct regarding the nature of principal ideals?

Would really appreciate this issue being clarified.

Peter

#### Klaas van Aarsen

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

===============================================================================================

"27.21 Definition

If R is a commutative ring with unity and [TEX] a \in R [/TEX] , the ideal [TEX] \{ ra | r \in R \} [/TEX] 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 [TEX] a \in R [/TEX]

=================================================================================================

Consider [TEX] N =\{ ra | r \in R \} [/TEX] ...........................(1)

If we take r = a in (1) then we have [TEX] ra = aa = a^2 \in N [/TEX]

If we take r = a and [TEX] a^2 \in N [/TEX] the we have using (1) again that [TEX] ra = a^2 a = a^3 \in N[/TEX]

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

Is the above analysis correct regarding the nature of principal ideals?

Would really appreciate this issue being clarified.

Peter
Hi Peter!

Yes, that is correct.

You might also say that $r=a^n \in R$ for $n \in \mathbb N$, so $a^n \cdot a = a^{n+1} \in N$.
And since you also have unity in R, it follows that $1a \in N$, and therefore $a^n \in N$.

#### Peter

##### Well-known member
MHB Site Helper
Thanks so much for that clarification - can now proceed on with some confidence