ar X iv : m at h / 04 10 22 6 v 3 [ m at h . R A ] 1 7 A ug 2 00 5 BRANCH RINGS , THINNED RINGS , TREE ENVELOPING RINGS

We develop the theory of “branch algebras”, which are infinitedimensional associative algebras that are isomorphic, up to taking subrings of finite codimension, to a matrix ring over themselves. The main examples come from groups acting on trees. In particular, for every field k we construct a k-algebra K which • is finitely generated and infinite-dimensional, but has only finite-dimensional quotients; • has a subalgebra of finite codimension, isomorphic to M2(K); • is prime; • has quadratic growth, and therefore Gelfand-Kirillov dimension 2; • is recursively presented; • satisfies no identity; • contains a transcendental, invertible element; • is semiprimitive if k has characteristic 6= 2; • is graded if k has characteristic 2; • is primitive if k is a non-algebraic extension of F2; • is graded nil and Jacobson radical if k is an algebraic extension of F2.