Abstract commutative ideal theory

Several years ago, M. Ward and the author [4] began a study in abstract form of the ideal theory of commutative rings. Since it was intended that the treatment should be purely ideal-theoretic, the system which was chosen for the study was a lattice with a commutative multiplication. For such multiplicative lattices, analogues of the Noether decomposition theorems for commutative rings were formulated and proved. However the theorems corresponding to the deeper results on the ideal structure of commutative rings were not obtained; the essential difficulty being the problem of formulating abstractly the notion of a principal ideal. This difficulty occurred in a mild form in treating the Noether theorem on decompositions into primary ideals. In fact, in the above mentioned paper, a weak concept of “principal element” was introduced which sufficed for the proof of the decomposition theorem into primaries. Nevertheless, the definition had serious defects and it was immediately obvious that it was not adequate for the further development of the abstract theory.