A radical for lattice-ordered rings

Lattice-Ordered Rings and Function Rings

Introduction: This paper treats the structure of those lattice-ordered rings which are subdirect sums of totally ordered rings—the f-rings of Birkhoff and Pierce [4]. Broadly, it splits into two parts, concerned respectively with identical equations and with ideal structure; but there is an important overlap at the beginning. D. G. Johnson has shown [9] that not every /-ring is unitable, i.e. e...

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 th...

On fuzzy convex lattice-ordered subgroups

In this paper, the concept of fuzzy convex subgroup (resp. fuzzy convex lattice-ordered subgroup) of an ordered group (resp. lattice-ordered group) is introduced and some properties, characterizations and related results are given. Also, the fuzzy convex subgroup (resp. fuzzy convex lattice-ordered subgroup) generated by a fuzzy subgroup (resp. fuzzy subsemigroup) is characterized. Furthermore,...

Ordered Rings and Fields

Radical Extensions of Rings

Jacobson's generalization [5, Theorem 8] of Wedderburn's theorem [8] states that an algebraic division algebra over a finite field is commutative. These algebras have the property that some power2 of each element lies in the center. Kaplansky observed in [7] that any division ring, or, more generally, any semisimple ring, in which some power of each element lies in the center is commutative. Ka...