Branch Rings, Thinned Rings, Tree Enveloping Rings
  We develop the theory of ``branch algebras'', which are infinite-dimensional 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 (1) is finitely generated and infinite-dimensional, but has only finite-dimensional quotients;   (2) has a subalgebra of finite codimension, isomorphic to $M_2(K)$;   (3) is prime;   (4) has quadratic growth, and therefore Gelfand-Kirillov dimension 2;   (5) is recursively presented;   (6) satisfies no identity;   (7) contains a transcendental, invertible element;   (8) is semiprimitive if k has characteristic $\neq2$;   (9) is graded if k has characteristic 2;   (10) is primitive if k is a non-algebraic extension of GF(2);   (11) is graded nil and Jacobson radical if k is an algebraic extension of GF(2). 
Laurent Bartholdi