The 𝒥-Radical and the Pierce Radicals of a Lattice-Ordered Ring

Abstract

The 𝒥-radical of a lattice-ordered ring Ris the ℓ-ring analogue of the Jacobson radical of a ring. It is shown that if the matrix ring R _n has the usual lattice order, then 𝒥(R _n ) = 𝒥(R) _n . The connection between an element abeing right ℓ-quasi-regular and the inequality a ○ x ≤ 0 is also investigated. For squares in an f-ring the connection is an equivalence. In general it is still an equivalence provided xis the sum of elements from a larger f-ring whose absolute values lie in R. It is also shown that the vanishing of annihilators in an f-ring is inherited by enough totally ordered homomorphic images to give a subdirect product decomposition.

Key Words:

• Lattice-ordered ring
• Radical
• 1-quasi-regularity
• Matrix 1-ring
• Subdirect product decompositions

Acknowledgments