  We will use commutators to provide decompositions of $3\times 3$ matrices as
sums whose terms satisfy some polynomial identities, and we apply them to
bounded linear operators and endomorphisms of free modules of infinite rank. In
particular it is proved that every bounded operator of an infinite dimensional
complex Hilbert space is a sum of four automorphisms of order $3$ and that
every simple ring that is obtained as a quotient of the endomorphism ring of an
infinitely dimensional vector space modulo its maximal ideal is a sum of three
nilpotent subrings.

Rings and Algebras