If A and B are ideals of a commutative ring R with unity and A+B=R show that A nn B = AB

boitshupoO 2020-12-09 Answered
If A and B are ideals of a commutative ring R with unity and A+B=R show that AB=AB
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Expert Answer

avortarF
Answered 2020-12-10 Author has 113 answers

The set A and B should satisfy these two conditions,
1. ABAB
2. ABAB
For xAB,
xAandxB
From problem, A and B are ideal of a commutative ring R with unity,
1R, and A+B=R
So,
For element aAandbB,
a+b=1
As xAandxB,
x=1
(1)x=1
(a+b)x=1
ax+bx=1
ax+xb=1 (R is commutative)
For ax, aAandxBandforxb,xAandbB, then
ax+xb=1 B
This indicates that,
ABAB
As A and B are ideal, then for any yBandzA, the product yzA.
For xAB,
x=i=1naibi
For, n>0, here, aiAandb1B.
So, the product aibiA
This indicates that,
x=i=1naibiAB
So, ABAB
Here, both conditions are satisfied.

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