Chain conditions on convolution rings

The notion of a convolution type has been introduced earlier. Imposing such a type on a ring gives the corresponding convolution ring. Under this umbrella, a wide variety of ring constructions can be covered, including polynomials, matrices, incidence algebras, necklace rings, group rings, and quaternion rings. Here conditions on the parameters of the convolution type to ensure that the corresponding convolution rings enjoy certain algebraic properties will be investigated. In particular, the requirements for the convolution rings to satisfy chain conditions are determined.