Hereditary ℓ-ideals of matrix rings over ℓ-rings

ABSTRACT

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

KEYWORDS:

• ℓ-Ring
• ℓ-ideal
• hereditary ℓ-ideal of a matrix ring
• strong absorbing property

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 06F25
• 13J25

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The authors acknowledge the support of NSFC [grant 11771004], and National Undergraduate Training Program for Innovation and Entrepreneurship [grant 201610006074].