Hereditary ℓ-ideals of matrix rings over ℓ-rings. Issue 12 (2nd December 2019) - Bai, Ruiqi
ABSTRACT: 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.