Every linear transformation is a sum of nonsingular ones

Abstract

In noncommutative Galois theory, the problem arises2 as to whether the ring of all linear transformations on a vector space over a division ring is generated by its units (=the nonsingular linear transformations=the automorphisms of the space). Equivalently, is every singular linear transformation a sum of nonsingular ones? The answer is trivially yes when the space is finite-dimensional and the division ring is an infinite (commutative) field, for then it suffices to choose, for each linear transformation a, a nonzero scalar a not a characteristic root of a, and write3 a =at+(a-a). The proof is not much more difficult in the general finite-dimensional case (cf. Case I in the proof of the theorem below). In this note we show that in the absence of restrictions on the vector space, the answer is still yes and in fact that every linear transformation is still a sum of two nonsingular ones (with the trivial exception of the identity transformation on a space of two elements). We use the following notations: ,N for O is an isomorphism of the vector space M onto the vector space All isomorphisms will be isomorphisms onto. Mappings will be written on the right: xa for the image of x under the mapping a; and hence af3 for the mapping obtained by applying first a and then d. M=L OK will have its usual significance: M is the sum of its complementary subspaces L and K. If a and 3 are linear mappings of L and K into a space N, a GO will denote the unique linear mapping of M into N which induces a on L and 3 on K. Note that if a and 3 are isomorphisms onto complementary subspaces of N, then a e3 is an isomorphism of M onto N. Notation for direct sum of a collection of subspaces or mappings: EMi, Eai.