Matrix rings having the intesection property of quasi-ideals

Abstract

Let R be a ring. For subsets A and B of R, let AB denote the set of all finite sums of the form sigma n i=1 aibi where aiEA and biEB. An additive subgroup Q of R is said to be a quasi-ideal of R if RQ intersection QR Q. A quasi-ideal Q of R is said to have the intersection property if there exist a left ideal H and a right ideal K of R such that Q = H intersection K. If each quasi-ideal of R has the intersection property, we say that R has the intersection property of quasi-ideals. For a positive integer n, let Mn(R) and SUn(R) denote the full nxn matrix ring over R and the ring of all strictly upper triangular nxn matrices over R, respectively. For a positive integer m, let Zm denote the ring of integers modulo m. The main results of this research are as follows. Theorem 1. Let R be a ring with identity, R>1 and char(R) is not equal to 2. If n is a positive integer such that SUn(R) has the intersection property of quasi-ideals, then n is less than or equal to 3. Theorem 2. If R is a division ring, then every quasi-ideal of SU3(R) is a left ideal or a right ideal. Hence for every division ring R, SU3(R) has the intersection property of quasi-dieals. Theorem 3. Let k be a positive integer and p a prime. Then every quasi-ideal of SU3(Zpk) is a left ideal or a right ideal. Hence SU3(Zpk) has the intersection property of quasi-ideals. Theorem 4. Let n and k be positive integers and p a prime. Then the following statements hold. (1) If p>2, then Mn(kZ2P) has the intersection property of quasi-ideals.