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The Nature of Principal Ideals

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,918
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
Mar 5, 2012
8,780
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
Jun 22, 2012
2,918
Thanks so much for that clarification - can now proceed on with some confidence :D