When is every linear transformation a sum of two commuting invertible ones?

A well-known result of Wolfson [7] and Zelinsky [8] says that every linear transformation of a vector space V over a division ring D is a sum of two invertible linear transformations except when dim(V)=1 and D=F2. Indeed, many of these linear transformations satisfy a stronger property that they are sums of two commuting invertible linear transformations. The goal of this note is to prove that every linear transformation of a vector space V over a division ring D is a sum of two commuting invertible ones if and only if |D|⩾3 and dim(V)<∞. As a consequence, a sufficient and necessary condition is obtained for a semisimple module to have the property that every endomorphism is a sum of two commuting automorphisms.