Let R be the ring of linear transformations of a right vector space over a division ring D. Three results are proved: (1) if |D|>4, then for any a∈R there exists a unit u of R such that a+u,a−u and a−u−1 are units of R; (2) if |D|>3 , then for any a∈R there exists a unit u of R such that both a+u and a−u−1 are units of R; (3) if |D|>2 , then for any a∈R there exists a unit u of R such that both a−u and a−u−1 are units of R. The second result extends the main result in H. Chen, [‘Decompositions of countable linear transformations’, Glasg. Math. J. (2010), doi:10.1017/S0017089510000121] and the third gives an affirmative answer to the question raised in the same paper.
