Hereditary ℓ-ideals of matrix rings over ℓ-rings. Issue 12 (2nd December 2019)

• Record Type: Journal Article
• Authors: Bai, Ruiqi; Yang, Yichuan
• Is Part Of: Linear & multilinear algebra. Volume 67:Issue 12(2019)
• Page Start: 2540
• Page End: 2548
• Publication Date: 2019-12-02
• Subjects: ℓ-Ring -- ℓ-ideal -- hereditary ℓ-ideal of a matrix ring -- strong absorbing property; 06F25 -- 13J25; Algebras, Linear -- Periodicals; Multilinear algebra -- Periodicals; 512.505
• DOI: 10.1080/03081087.2018.1497583
• Languages: English
• ISSNs: 0308-1087

Abstract

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.