Link between a natural centralizer and the smallest essential ideal in structural matrix rings

Abstract

In a structural matrix ring M_n (ρ R) over an arbitrary ring R we determine the centralizer of the set of matrix units in M_n (ρR) associated with the anti-symmetric part of the reflexive and transitive binary relation ρ on {1,2,…,n}. If the underlying ring R has no proper essential ideal, for example if R is a field, then we show that the largest ideal of M_n (ρR) contained in the mentioned centralizer coincides with the smallest essential ideal of M_n (ρR).

Keywords:

• Structural matrix ring
• centralizer
• esential ideal

1991 Mathematics Subject Clasification:

• Primary 16S50
• Primary 16D25