h-ideals and l-prime l-ideals in lattice-ordered rings
by
Jingjing Ma
University of Houston Clear Lake

Let R be a commutative ring with an identity element. Recall that an ideal I of R is called a z-ideal if whenever a, b are contained in the same set of maximal ideals and a is in I, then b is in I. This useful concept was used in studying the ideal structure of the ring C(X) of continuous real-valued functions over a topological space X. In his 1973 paper "z-ideals and prime ideals", G. Mason has generalized this concept to general rings, and used it to study the ideal structure of f-rings. Since then many work has been done by M. Henriksen, C. B. Huijsmans, S. Larson, B. de Pagter, F. A. Smith, and many others.

Now let A be a commutative lattice-ordered ring with a positive identity element. If we change "maximal ideal" to "maximal l-ideal" in the above definition of a z-ideal, then we have the definition of an h-ideal. h-ideals were first introduced by H. Subramanian in his 1967 paper "l-prime ideals in f-rings" to study the ideal structure of a commutative f-ring with an identity element. He noticed that some results for C(X) can be generalized to commutative f-rings with identity elements.

We notice that some results obtained in H. Subramanian's paper are actually true for a commutative lattice-ordered ring with a positive identity element. In the present talk, we study the l-ideal structure of a lattice-ordered ring with a positive identity element by using h-ideals.

Date received: December 17, 2002