Minimal quasi-ideals of some matrix rings

Abstract

Let R be a ring. An additive subgroup Q of a ring R is said to be a quasi-ideal of R if RQ intersection QR Q. For a R, let (a)q denote the quasi-ideal of R generated by a. A quasi-ideal Q of R is said to be minimal if Q is not equal to {0} and Q does not properly contain any nonzero quasi-ideal of R. Therefore if Q is a minimal quasi-ideal of R, then Q = (a)q for every a Q\{0}. Let F be a field, n a positive integer, k {1,2,...,n} and Mn(F) = the full nxn matrix ring over F, SUn(F) = the ring of all strictly upper triangular nxn matrices over F, C2n+1(F) = the ring of all (2n+1)x(2n+1) matrices A over F with Aij = 0 for all (i,j) {1,2,...,2n+1}x{1,2,...,2n+1}\(1,1),(1,2n+1),(n+1,n+1),(2n+1,1),(2n+1,2n+1)} and Rn(F,K) = the ring of all nxn matrices A over F with Aij = 0 for all i,j {1,2,...,n} and i is not equal to k. The main results of this research are as follows: Theorem 1. For A Mn(F),(A)q is a minimal quasi-ideal of Mn(F) if and only if rank(A) =1. Theorem 2. If char(F) = 0, then SUn(F) has no minimal quasi-ideal. Theorem 3. Let char(F) = p>0. 1) For A SUn(F), if rank(A) = 1, then (A)q is a minimal quasi-ideal of SUn(F). 2) The converse of 1) holds if and only if n<3. Theorem 4. For A C2n_1(F), (A)q is a minimal quasi-ideal of C2n+1(F) if and only if rank(A) = 1. Theorem 5. Let char(F) = 0 and A Rn(F,k). Then (A)q is a minimal quasi-ideal of Rn(F,k) if and only if Akk is not equal to 0. Theorem 6. If char(F) = p>0, then for any A Rn(F,k), (A)q is a minimal quasi-ideal of Rn(F,k)