How Do Ideals Generated by Powers of an Element Relate in Commutative Rings?

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A^rm-1)((a^m)-1) + ... + (a^d)((a^n)-1) = = (a^d)((a^m)-1) + ... + (a^d)((a^n)-1) = = (a^d)((a^m)-1 + ... + (a^n)-1) = = (a^d)(a^gcd(m,n)-1) = = (a^gcd(m,n)d)-1 = = (a^gcd(m,n))-1.In summary, the conversation discusses how to show that the ideal generated by (a^m)-1 and (a^n)-1 is equal to (a^
  • #1
joecoz88
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Hello,

Let [a,b] be the ideal generated by a and b.

If R is a commutative ring with unity, let a be in R and m,n be natural numbers.

Show that [ (a^m)-1, (a^n)-1 ] = [ (a^gcd(m,n)) -1 ]


Seems simple but I am having trouble with it. Thanks in advance!
 
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  • #2
LHS included in RHS:
Let d=gcd(m,n), m=ud, n=vd. Apply the formula
(x-1)(x^(k-1)+x^(k-2)+...+x+1)=x^k-1
to x=a^d, k=u,v.

RHS included in LHS:
Let d=gcd(m,n)=rm-sn with positive r,s and A=a^m, B=a^n
(a^d)-1 = (A^r)-1 - (a^d)((B^s)-1) =
= (A^(r-1)+...+A+1)((a^m)-1) - (a^d)(B^(s-1)+...+B+1)((a^n)-1)
 
Last edited:

FAQ: How Do Ideals Generated by Powers of an Element Relate in Commutative Rings?

What is a finitely generated ideal?

A finitely generated ideal is a subset of a ring that can be generated by a finite number of elements. In other words, it is the smallest ideal that contains a finite set of elements.

How is a finitely generated ideal different from an ideal?

A finitely generated ideal is a specific type of ideal that can be generated by a finite set of elements. In contrast, an ideal may or may not be able to be generated by a finite set of elements.

What is the significance of finitely generated ideals in mathematics?

Finitely generated ideals play an important role in abstract algebra and commutative algebra. They have applications in algebraic geometry, number theory, and coding theory.

Can all ideals be finitely generated?

No, not all ideals can be finitely generated. For example, the ideal of all polynomials with integer coefficients is not finitely generated.

How are finitely generated ideals used in practical applications?

Finitely generated ideals have applications in fields such as computer science and engineering, particularly in the design and implementation of algorithms for solving linear equations and systems. They also have applications in cryptography and error-correcting codes.

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