Representable distributive Noether lattices
Eric T. Johnson, J. P. Lediaev · Pacific Journal of Mathematics · 1969
Recently, Bogart showed that a certain class of distributive Noether lattices, namely regular local ones, are embeddable in the lattice of ideals of an appropriate Noetherian ring.In this paper a characterization of the distributive Noether lattices which are representable as the complete lattice of ideals of a Noetherian ring is obtained.We observe that if L(R) is the lattice of ideals of a ring R (commutative with 1) and if A, B and C are elements of L(R) with A ^ B and A ^ C, then there exists a principal element E e L(R) with E ^ A, E ^ B and E ^ C. If a Noether lattice L has this property, then we will say that L satisfies the weak union condition.(The term union condition has been used elsewhere for a stronger property.)With this definition, then, the main result of this paper is that a distributive Noether lattice L is representable as the lattice of ideals of a Noetherian ring if, and only if, L satisfies the weak union condition.We adopt the terminology of [2] and we assume throughout that L is a Noether lattice.LEMMA 0. // L is local, and if the maximal element PeL is principal, then every element A Φ 0 of L is a power P n (0 rg n) of P.