On a Class of Lattice-ordered Rings

for some real number X, the symbol V denoting the lattice least upper bound. Any ring R is regular [10] if for each xER there is an xaER such that xx°x = x. It is evident that every regular F-ring R contains a maximal bounded sub-F-ring R, the F-ring of all xER satisfying equation (1.1). The relationship between a regular F-ring and its maximal bounded sub-F-ring is analogous to that between the ring of all continuous functions on a completely regular space X and the ring of all bounded continuous functions on X. For example, it is shown in Theorem 3 that there is a one-to-one correspondence between the maximal ideals of R and those of R. (For the theory of rings of continuous functions, see [5] and [6].) A maximal ideal M of a ring R is real [6] if the quotient ring R — M is ring-isomorphic to the real field. An ideal 5 of an F-ring R is closed if anES, n=l and V^-i a„Gi? imply V„°.i «n£5. It is proved in Theorems 5 and 6 that the closed maximal ideals of a regular F-ring are real and that there is a one-to-one correspondence between the closed maximal ideals of a regular F-ring R and the closed maximal ideals of R, the maximal bounded sub-F-ring of R. It is a direct corollary of some results of Nakano [9, pp. 39, 212] that a bounded F-ring is ringand lattice-isomorphic to the ring of all continuous functions on a compact Hausdorff space. Therefore every bounded F-ring is a semisimple real Banach algebra. "Real" is used here in the classical sense, that is, a partially ordered ring R is