Generalized matrix rings having the intersection property of quasi-ideals

Abstract

Let R be a ring. For nonempty subsets A,B R, let AB denote the set of all finite sums of the form sigmaaibi where ai E A and bi E B. A subring Q of R is called a quasi-ideal of R if RQ QR Q. It is known that the intersection of a left ideal and a right ideal of R is a quasi-ideal but a quasi-ideal of R need not be obtained in this way. The ring R is said to have the intersection property of quasi-ideals if every quasi-ideal of R is the intersection of a left ideal and a right and a right ideal of R. In the remainder, let R be a division ring and m and n positive integers. We denote by Mm,n(R) the set of all m x n matrices over R. For P Mn,m(R), let(Mm,n(R), +, P) be the ring Mm,n(R) under usual addition and the multiplication * defined by A * B = APB for all A,B Mm,n(R). Let Mn,n(R)=Mn(R) and we denote by SUn(R) the set of all strictly upper triangular matrices in Mn(R). For an upper triangular n x n matrix P over R, let (SUn(R), +, P) be defined similarly. The main results of this research are as follows: Theorem 1. For P Mn,m(R), the ring (Mm,n(R), +, P) has the intersection property of quasi-ideals if and only if either P = 0 or rank P = min {m, n}. Corollary 2. For P Mn(R), the ring (Mn(R), +, P) has the intersection property of quasi-ideals if and only if either P = 0 or P is invertible. Theorem 3. For an upper triangular n x n matrix P over R, the ring (SUn(R), +, P) has the intersection property of quasi-ideals if and only if one of the following statements holds. (i) n<3. (ii) n=4 and P22=0 or P33=0. (iii) n>4, Pij=0 for all i,j {3, 4,...,n-2} and (a) P2j=0 for all j {2, 3,...,n-2} or (b) Pi,n-1=0 for all i {3, 4,...,n-1}.