Simple ring/ideal query

  1. 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.
  2. jcsd

  3. 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...

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