Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 1 / 9E (Groups, Rings and Modules)

Assume (i). For $a,b\in R$, write the ideal $(a,b)=(d)$. Then $d=ra+sb$ for some $r,s\in R$. Since $a,b\in (d)$, the element $d$ divides both; and every common divisor of $a,b$ divides the displayed linear combination $d$. Thus $d$ is a greatest common divisor and (ii) holds.

Assume (ii), and let an $R$-submodule $M\subseteq F$ be generated by $x_1,\dots ,x_m$. Choose a common nonzero denominator $d$ and write $x_j=a_j/d$ with $a_j\in R$. Repeated application of (ii) shows thatfor one element $g$: the gcd at each step is a linear combination of the elements processed so far. Thereforeso (iii) holds. This is the finite-ideal property defining a Bézout domain.

Assume (iii). Because $R$ is a Noetherian ring, every ideal $I\subseteq R$ is finitely generated. Regard $I$ as an $R$-submodule of $F$. By (iii), $I=Rx$ for some $x\in F$; since $x\in I\subseteq R$ unless $I=0$, this is a principal ideal of $R$. Thus every ideal is principal and (i) holds.

Together with the previous two implications, this proves (i), (ii), and (iii) equivalent.

If (i) holds, the submodule theorem for free modules over a principal ideal domain says that any submodule $N\subseteq R^n$ is free. Its rank is at most $n$, since tensoring the inclusion with $F$ embeds $N{\otimes }_{R}F$ into $F^n$. Hence $N$ has at most $n$ generators, proving (iv).

Conversely, apply (iv) with $n=1$. Every ideal is an $R$-submodule of $R$ and therefore has one generator, which is (i). Thus all four conditions are equivalent.

Suppose the additive group of the integral domain $R$ is isomorphic to ${\mathbb{Z}}^n$. Every ideal $I$ is then an additive subgroup of a finitely generated free abelian group, so it has finitely many additive generators $x_1,\dots ,x_m$. These also generate $I$ as an $R$-ideal: integer coefficients are coefficients from the canonical copy of $\mathbb{Z}$ in $R$. Thus every ideal is finitely generated, proving that $R$ is Noetherian by noetherianity from finite additive rank.

For an example that fails (i)--(iv), takeIt is an integral domain and has additive group ${\mathbb{Z}}^2$. The idealhas index two: modulo $I$, one has $2=0$ and $\sqrt{-5}=-1=1$, and the resulting quotient is ${\mathbb{F}}_2$. If $I=(a+b\sqrt{-5})$, multiplication by its generator would have determinant and absolute indexwhich has no integer solution. Hence $I$ is not principal, as detailed in nonprincipal ideal in the integers adjoined a square root of minus five. Condition (i), and therefore all four equivalent conditions, fails.