a principal ideal contains a monic polynomial of least degree n

Question

Q11.3.11 Artin Algebra 2nd

Let $R$ be a ring, and let $I$ be an ideal of the polynomial ring $R[x]$. Let $n$ be the lowest degree among nonzero elements of $I$. Prove or disprove the following: $I$ contains a monic polynomial of degree $n$ if and only if it is a principal ideal.

I've proved the first direction, now I want to prove:

a principal ideal contains a monic polynomial of least degree $n$

Is it trivial? I kind of missing the starting point, and can not find any resources on this, any help?

Thanks in advance~

Answer 1

If you are not working over a field, then principal ideals of $R[X]$ may not contain any monic polynomials at all, let alone such polynomials of minimal degree. Principal ideals with a monic generator$~P$ obviously do have such an element (namely $P$, which is monic and clearly of minimal degree for a nonzero multiple of$~P$). But these are in fact all cases where it works. A simple example of failure is the ideal generated by $91X^3$ in $\Bbb Z[X]$ (or by $2X$ if you prefer small examples).

My "these are all cases" claim is just the implication: "if $I$ has a monic polynomial$~P$ among its nonzero elements of minimal degree, then $I$ is the principal ideal generated by$~P$", which you say you have already shown (presumably using Euclidean division by$~P$). Note that not all generators of$~I$ are monic (they could have invertible leading coefficient) nor even do they have to be of minimal degree if $R$ has zero divisors (if $n\in R$ is nilpotent then $1-nX^d\in R[X]$ is invertible for any $d>0$, so the ideal it generates is all of $R[X]$, with monic generator $1$ of degree $0<d$).

Answer 2

If $I$ is the ideal over integers mod 4 generated by $2x^2+2$ then $I$ has no monic polynomials of degree 2.

Note I'm indebted to Daniel Fisher's comment to choose mod 4 rather than mod 3 as I had earlier.

An element of the ideal $I$ is obtained by multiplying $2x^2+2$ either by a polynomial of degree at least one (which makes the degree 3 or more) or else by multiplying $2x^2+2$ by an element of the integers mod 4. Since $2$ has no inverse mod $4$ we cannot choose a constant so as to multiply $2x^2+2$ by it, and obtain a monic polynomial of degree 2. This shows that this $I$ has no monic polynomials of degree 2.

By the way I am taking $n$ as the lowest positive degree. If we allow degree zero, i.e. nonzero constants, as the generating polynomials, we can take the case of the single element $2$ in the polynomials over the integers mod $4$, and generate the ideal $I=\{0,2\}$ as an example, which is an ideal of only constant polynomials, and here $n=0$ and there is no monic polynomial of degree $0$ in $I$ since that monic would have to be $1$.