# Proper-class sized “ring” with no maximal ideals

Suppose I have a collection of "elements" together with operations that satisfy the axioms for a commutative ring with identity --- except that these elements form not a set, but a proper class.

Must such a thing contain a maximal ideal (where an "ideal" is allowed to be proper-class-size)? Obviously, the usual Zorn's Lemma argument is not available.

Suppose $x_\alpha$, $\alpha \in \mathrm{Ord}$, enumerates the elements of the proper class ring $R$. Without loss of generality $x_0$ is the zero of the ring $R$. Define $h(0) = 0$ and for each $\alpha \gt 0$, let $h(\alpha)$ be the first ordinal $\eta$ (if any) such that $x_\eta$ is not in $I_\alpha = \sum_{\beta\lt\alpha} Rx_{h(\beta)}$ and $Rx_\eta + I_\alpha \neq R$. These $R$-ideals are all uniformly definable from $R$, $\langle x_\alpha\rangle_{\alpha\in\mathrm{Ord}}$ and $\langle h(\beta)\rangle_{\beta\lt\alpha}$, so there is no trouble carrying out this construction in NBG. The elements indexed by the function $h$ will enumerate a (not necessarily proper) class of generators for a maximal $R$-ideal.