On subrings of matrix rings over fields

Abstract

Let K be a commutative field and let V be an n-dimensional vector space over K. We denote by L(V) the ring of all K-linear endomorphisms of V into itself. A subring of L(V) is always assumed to contain the unit element of L(V), but it need not be a vector subspace of the K-algebra L(V). Suppose now that A is a subring of L(V). Then we may consider L(V) as a left or a right A-module. We may also consider V as a left A-module, i.e. if x ∈ V and if f ∈ A then we get the image element f(x) in V. If we choose a basis for V over K then we get the associated matrix representation of L(V). So in this way L(V) is identified with the full matrix ring Mn(K), and in this way the study of subrings of L(V) can be reduced to the study of subrings of Mn(K).