Minimal Quasi-ideals and Minimal Bi-ideals of Generalized Rings of Strictly Upper Triangular Matrices

Abstract: A quasi-ideal of a ring R is a subring Q of R such that RQ∩QR ⊆ Q where RQ [QR] is the set of all finite sums of the form riqi [ qiri] with ri ∈ R and qi ∈ Q, and by a bi-ideal of R we mean a subring B of R such that BRB ⊆ B. Let SUn(F) be the set of all strictly upper triangular n × n matrices over a field F. For an upper triangular n × n matrix P over F, let (SUn(F),+, P) denote the ring SUn(F) under the usual addition of matrices and the multiplication ∗ defined by A ∗ B = APB for all A,B ∈ SUn(F). Here we determine minimal quasi-ideals and minimal bi-ideals of the ring (SUn(F),+, P). It is also shown that if n> 1 and charF 6 = 0, then a minimal quasi-ideal and a minimal bi-ideal of (SUn(F),+, P) always exist, and it has neither minimal quasi-ideals nor minimal bi-ideals if charF = 0