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

in a structural matrix ring M-n (rho, R) over an arbitrary ring R we determ ine the centralizer of the set of matrix units in M-n (rho, R) associated w ith the anti-symmetric part of the reflexive and transitive binary relation rho on {1, 2, ..., n}. if the underlying ring R has no proper essential id eal, for example if R is a field, then we show that the largest ideal of M- n (rho, R) contained in the mentioned centralizer coincides with the smalle st essential ideal of M-n (rho, R).