Calyxes and Corollas

A calyx (multiplicative lattice) is a complete lattice endowed with the structure of a monoid such that multiplication by an element is a left adjoint functor of complete lattices (equivalently, a left adjoint functor which preserves colimits). Calyxes are a generalization of the set of ideals of a ring, which form a complete lattice under intersection and summation; in conjunction with the natural multiplication operation (ideal multiplication), this lattice forms a calyx. One can axiomatize the submodules of a module similarly to arrive at the definition of a corolla (module lattice). We present here a notion of principal element equivalent to Dilworth’s original definition of principal on a multiplicative lattice, and show that this is equivalent to weak principal. The new concept of principal element begins with a notion of morphism in the category of corollas over a given calyx. We characterize principal elements in the calyx induced by a Noetherian ring as those ideals in the lattice which are principal ideals (in the ordinary sense) after localizing at any given prime ideal. We include here w