Maximal Left Ideals and Idealizers in Matrix Rings

Stone, David R.

Canadian journal of mathematics, Tome 32 (1980) no. 6, pp. 1397-1410

Résumé

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 → M n(A) to form the lattice of two-sided ideals of Mn(R). We wish to study the more complex left ideal structure of M n(R). For example, if K is a commutative field, then M n(K) has non-trivial left ideals. In particular M n(K) has the maximal left ideal consisting of all matrices with some designated column zero. Or for any ring with maximal left ideal M, M n(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 M n(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 M n(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 M n(K) can be naturally identified with P n–1(K) (projective space).

Détail

DOI : 10.4153/CJM-1980-110-8

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