On direct decompositions. II

Extract

The theory developed in our first paper is applied here to the case of algebras A such that every pair of congruences on A commute. The general refinement theorem which we obtain is already known when the ascending and descending chain conditions hold for the lattice of congruences. This result is due to O. Ore. We here weaken this finiteness assumption to a point comparable with that of group theory when both chain conditions are assumed for subgroups of the centre. Our path to the theorem is obtained by adapting to a different situation methods developed by Kurosch, Baer, and Jónsson and Tarski. From Kurosch (4) we take the concept of the centre of a pair of direct decompositions, which we extend and formulate as a pair of mappings π [α, β; γ, δ] over the set of congruences. In this connexion, Theorems 12·5 and 12·6 appear to be new, even for groups. From Baer (1) we take the concept of canonical refinements and generalize it to our case. The final approach (§ 14) to the refinement theorem is essentially following Jónsson-Tarski (3). Possibly our most interesting conclusion is that the elegant methods using endomorphisms, introduced by Fitting into group theory, generalize naturally to our case.