Maximal Left Ideals and Idealizers in Matrix Rings

Abstract

Let R be a ring with identity, Mn(R) the ring of n × n matrices over R. The lattice of two-sided ideals of R is carried via A → Mn(A) to form the lattice of two-sided ideals of Mn(R). We wish to study the more complex left ideal structure of Mn(R). For example, if K is a commutative field, then Mn(K) has non-trivial left ideals. In particular Mn(K) has the maximal left ideal consisting of all matrices with some designated column zero. Or for any ring with maximal left ideal M, Mn(R) has the maximal left ideal consisting of all matrices with some column's entries from M. In Theorem 1.2 we characterize the maximal left ideals of Mn(R) in terms of those of R. We briefly study some contraction properties of maximal left ideals in matrix rings. For R commutative we “count” the maximal left ideals of Mn(R) and describe the idealizer of any such ideal; in the case where K is a field we see that the collection of maximal left ideals of Mn(K) can be naturally identified with Pn–1(K) (projective space).