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

[joecoz88]

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!

 

[yyat]

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)