In this paper we determine the structure of joinprincipal elements in Noether lattices and we apply these results to obtain the Krull Principal Ideal Theorem for join-principal elements, a representation theorem for a class of Noether lattices, and some interesting ring results. The concept of a Noether lattice, introduced by R. P. Dilworth [1], is an abstraction of the lattice of ideals L of a Noetherian ring R. Fundamental to this abstraction is the following lattice-theoretic characterization of two key properties of a principal ideal E: (AAB:E)E=AEAB and (A VBE): E= (A: E)vB for all A, B e L. An element E of a multiplicative lattice Y is said to be a meet-principal (join-principal) element if it satisfies the first (second) identity. E is principal if it satisfies both identities. The concepts of meetand join-principalness have been instrumental in the generalization of classical results of the ideal theory of Noetherian rings to lattices ([1], [2], and [4]), and have provided a basis for representation and embedding theorems for certain classes of Noether lattices ([3], [4], and [5]). The results of [3], [4], and [5] hint that, in general, the connection between meetand join-principal elements in a Noether lattice may be quite close. In this paper, we obtain a structural result for join-principal elements (Theorem 1) which not only allows us to explore the relationship between meetand join-principal elements but it also yields as corollaries the Krull Principal Ideal Theorem for joinprincipal elements, the representation theorem in [4], the result that joinprincipal maximal elements have minimal bases consisting of independent elements, and some interesting ring theoretic results. We adopt the terminology of [1]. Let Y be a noether lattice and let M and E be elements of Y such that E is meet-principal and M is maximal. Then {E} is meet-principal, and therefore join-irreducible, in Y_IrIt follows that {E} is principal in Y_12, Received by the editors June 1, 1971. AMS 1970 subject classifications. Primary 06A20, 13A15; Secondary 13F05. Key X ords and phrases. Noether lattice, join-principal element, meet-principal element, principal element, Krull Principal Ideal Theorem, Noetherian ring, general ZPI-ring, Dedekind domain, weak union condition. ( American Mathematical Society 1972