A universal Krull–Lindenbaum theorem

We formulate a natural common generalisation of Krull's theorem on prime ideals and of Lindenbaum's lemma on complete consistent theories; this has instantiations in diverse branches of algebra, such as the Artin–Schreier theorem. Following Scott we put the Krull–Lindenbaum theorem in universal rather than existential form, which move allows us to give a relatively direct proof with Raoult's Open Induction in place of Zorn's Lemma. By reduction to the corresponding theorem on irreducible ideals that is due to Noether, McCoy, Fuchs and Schmidt, we further shed light on why prime ideals occur together with transfinite methods.