For a ring R with identity 1, a
preprime is a nonempty subset T of R which is closed under the two binary
operations, addition and multiplication, of R and with −1∉T. A prime of R is a
preprime of R which is maximal with respect to set inclusion. A field K is locally
finite if every member of K is a member of some finite subfield of K. For a finite
dimensional vector space V over K let G=HomK(V,V ) denote the full ring of
linear transformations of V over K. Let W and L be subspaces of V with
W ⊂ L ⊂ V aud W≠L. Let T(L,W) = {α ∈ G∕α ⋅ L ⊂ W}. Then T(L,W) is a
preprime of G. Let
We will show that the primes of G are exactly those preprimes T(L,W) ∈𝒯 with
dimKL = 1 +dimKW.
