THE INTERSECTION PROPERTY OF QUASI-IDEALS IN GENERALIZED RINGS OF STRICTLY UPPER TRIANGULAR MATRICES

Abstract

Let SUn(R) denote the ring of all strictly upper triangular n × n matrices over a division ring R. It is known that the ring SUn(R) has the intersection property of quasi-ideals if and only if n 3. In this paper, this result is generalized. Let P be an upper triangular n ×n matrix over R and (SUn(R),+,P) the ring SUn(R) under usual addition and the multiplication ∗ defined by A∗B = APB for all A,B ∈ SUn(R). We characterize when the ring (SUn(R),+,P) has the intersection property of quasi-ideals in terms of n and the entries of P. The above result then becomes a special case.