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The main result of this paper is the following: if F is any field and R isany F-subalgebra of the algebra of nxn matrices over F with Lie nilpotenceindex m, then the F-dimension of R is less or equal than M(m+1,n), whereM(m+1,n) is the maximum of a certain expression of m+1 nonnegative integers andn. The case m=1 reduces to a classical theorem of Schur (1905), latergeneralized by Jacobson (1944) to all fields, which asserts that if F is analgebraically closed field of characteristic zero, and R is any commutativeF-subalgebra of the full nxn matrix algebra over F, then the F-dimension of Ris less or equal than ((n^2)/4)+1. Examples constructed from block uppertriangular matrices show that the upper bound of M(m+1,n) cannot be lowered forany choice of m and n. An explicit formula for M(m+1,n) is also derived.

The maximum dimension of a Lie nilpotent subalgebra of M_{n}(F) of index m