On Defining Ideals or Subrings of Hall Algebras

Let A be a finitary algebra over a finite field k, and A-mod the category of finite dimensional left A-modules. Let H(A) be the corresponding Hall algebra, and for a positive integer r let Dr(A) be the subspace of H(A) which has a basis consisting of isomorphism classes of modules in A-mod with at least r+1 indecomposable direct summands. If A is hereditary of type An, then Dr(A) is known to be the kernel of the map from the twisted Hall algebra to the quantized Schur algebra indexed by n+ 1 and r. For any A, we determine necessary and sufficient conditions for Dr(A) to be an ideal and some conditions for Dr(A) to be a subring of H(A). For A the path algebra of a quiver, we also determine necessary and sufficient conditions for Dr(A) to be a subring of H(A).