Question

Let Z[X] be the polynomial ring with coefficients in Z.prove or disprove that the ideal I= (4,x) is a principal ideal in Z[X]. Is this ideal I a maximal ideal in Z[x]? Explain your answer

Answer 1

$I=(4,x)=4f_1+xf_2$ , $f_1,f_2\isin Z[x]$

An ideal is called principal if it can be generated by a single polynomial.

We have:

$I=\lang4,x\rang=\{a_nx_n+...+a_1+a_0;a_0\ is\ divisible\ by\ 4\}$

Auppose that $I=\lang f(x)\rang$ for some $f(x)\isin I$ .

If $f(x)$ is a constant polynomial, then $\lang f(x)\rang$ contains only polynomials with coefficients divisible by 4, and we do not get $x$ .

If $f(x)$ is of degree at least 1, then non-zero polynomials in $\lang f(x)\rang$ have degree at least 1, and we do not get 4.

So $I$ is not of the form $\lang f(x)\rang$, that is, $I$ is not a principal ideal.

.For the polynomial ring $Z[x]$ only ideal $\lang p,x\rang$ is maximal, where $p$ is a prime number.

So the ideal $I=\lang4,x\rang$ is not maximal.