Let $A=k[X], k$ a field. I am reading in a Algebra book that the ideal $(m)$ of all polynomials in $A$ with zero constant term is maximal. At the other hand, since $(m)$ is principal ideal, all polynomials in $(m)$ will have zero constant term. But having zero constant term means that $m$ is not irreducible ( one can factorize $X$ ) and thus not prime. Here I have a conflict since a maximal ideal is a prime one.
Can somebody help me understand what is here wrong ?
Thanks.
You can't factor an $X$ from the polynomial $X$ itself, though (well, you can, but not in the sense of the element being reducible). And $X\in k[X]$ is exactly the element which generates the maximal ideal of polynomials with zero constant term.