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