Hereditary ℓ-ideals of matrix rings over ℓ-rings

Let R be an ℓ-ring and let M_n(R) be the matrix ring over R. An ℓ-ideal I of M_n(R) is called hereditary if I = M_n(I) 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 M_n(R) (n ≥ 2) is hereditary? We first show that if R has the identity element 1 then all ℓ-ideals of M_n(R) 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.