ABSTRACT
Let R be an ℓ-ring and let be the matrix ring over R. An ℓ-ideal of is called hereditary if for some ℓ-ideal I of R. In this paper, we consider the following question: Which conditions on R determine that any ℓ-ideal of is hereditary? We first show that if R has the identity element 1 then all ℓ-ideals of are hereditary. It is natural to guess that the result also holds for arbitrary ℓ-rings. However, using infinitesimal continuous function rings, we construct counterexamples to show that it is not the case if R does not contain 1. Finally, we answer the question completely.
Disclosure statement
No potential conflict of interest was reported by the authors.