Structure of Noether lattices with join-principal maximal elements
E. Johnson, J. P. Lediaev · Pacific Journal of Mathematics · 1971
In this paper we explore the structure of Noether lattices with join-principal maximal elements.Results which completely specify the structure of certain special classes of Noether lattices, and relate them to lattices of ideals of Noetherian rings, have been obtained in and [8].For example, in [7] we showed that if every maximal element of a Noether lattice ^ is meet-principal, then £f is distributive and can be represented as the lattice of ideals of a Noetherian ring.Moreover, for distributive Noether lattices, the condition that every maximal element is meet-principal is equivalent to representability.In a more recent paper [8], we began considering the complementary case of a Noether lattice in which every maximal element is join-principal in order to determine the extent of the relationship between the two situations.There we showed that if 0 is prime in ^ (and every maximal element is join-principal), then £? is distributive and representable.Hence, if 0 is prime, the assumptions that every maximal element is meet-principal and that every maximal element is joinprincipal are equivalent, and either implies representability.In this paper, we continue the investigation begun in [8].Our results extend the class of Noether lattices for which embedding and structure theorems are known, and also introduce a construction process for Noether lattices which leads to new examples.In § 1, we show that in a local Noether lattice (£f, M) in which M is join-principal and not a prime of 0, the maximal element M has a minimal base E u ,E k of independent principal elements (i.e., E,A (E,V V Ei V V E k ) = 0 for i = 1, ..., k).And we use this result to show that if M is join-principal and not a prime of 0, then Sf is distributive.In § 2, we obtain structure and embedding theorems for distributive local Noether lattices with join-principal maximal elements.In § 3, we investigate some of the consequences of our results outside of the local case.We adopt the terminology of [5].