The Carlitz algebras

The Carlitz F_q-algebra C=C_ν, ν∈N, is generated over an algebraically closed field K (which contains a non-discrete locally compact field of positive characteristic p>0, i.e. K?F_q[[x,x⁻¹]], q=p^ν), by the (power of the) Frobenius map X=X_ν:f?f^q, and by the Carlitz derivativeY=Y_ν. It is proved that the Krull and global dimensions of C are 2, classifications of simple C-modules and ideals are given, there are only countably many ideals, they commute (IJ=JI), and each ideal is a unique product of maximal ones. It is a remarkable fact that any simple C-module is a sum of eigenspaces of the element YX (the set of eigenvalues for YX is given explicitly for each simple C-module). This fact is crucial in finding the group of F_q-algebra automorphisms of C and in proving that any two distinct Carlitz rings are not isomorphic (C_ν?C_μ if ν≠μ). The centre of C is found explicitly, it is a UFD that contains countably many elements.