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Simple ring/ideal query

by Ad123q
Tags: query, ring or ideal, simple
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Ad123q
#1
Apr7-12, 07:25 AM
P: 19
I have in my notes that (2X,5) is an ideal of Z[X], but I can't see why this can be so.

For example 5+2X is in (2X,5) and 7+X is in Z[X] but then

(5+2X)(7+X) =
= 35+5X+14X+2X^2
= 2X^2+19X+35.

19 is not divisible by 2 and so this element is not in (2X,5), contradicting the "absorbance" property of ideals.
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DonAntonio
#2
Apr7-12, 08:02 AM
P: 606
Quote Quote by Ad123q View Post
I have in my notes that (2X,5) is an ideal of Z[X], but I can't see why this can be so.

For example 5+2X is in (2X,5) and 7+X is in Z[X] but then

(5+2X)(7+X) =
= 35+5X+14X+2X^2
= 2X^2+19X+35.

19 is not divisible by 2 and so this element is not in (2X,5), contradicting the "absorbance" property of ideals.


You seem to believe that any element in [itex](2x,5)[/itex] must have an even lineal coefficient, but this is wrong: the 5 there can multiply some x-coeff. of some

pol. and added to the even coefficient in the other factor we get an odd coef.

For example, the element [itex]2x\cdot 1 + 5\cdot x = 7x[/itex] belongs to the ideal...

DonAntonio


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