I'm having trouble proving that the ideal generated by $ \langle x,x+2 \rangle $ in the polynomial ring over the integers is not principal. I think I understand the problem I'm just not sure how to actually prove it. If we have two first degree polynomials in ring of the form $(x+a),(x+b)$ then their product obviously has the form $(x^2+(a+b)x+ab)$ which obviously is equivalent to $x^2+bx$ if one of the constants is zero. I think that's the main property that I should focus on but I'm not sure of how I could progress, if someone could help I'd appreciate it.
2026-03-29 16:34:35.1774802075
An example of non-principal Ideal in polynomial ring over integers
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