The Carlitz algebras

The Carlitz FqFq-algebra C=CνC=Cν, ν∈Nν∈N, is generated over an algebraically closed field KK (which contains a non-discrete locally compact field of positive characteristic p>0p>0, i.e. K≃Fq[[x,x−1]]K≃Fq[[x,x−1]], q=pνq=pν), by the (power of the) Frobenius   map X=Xν:f↦fqX=Xν:f↦fq, and by the Carlitz derivative  Y=YνY=Yν. It is proved that the Krull and global dimensions of CC are 2, classifications of simple CC-modules and ideals are given, there are only countably many   ideals, they commute (IJ=JI)(IJ=JI), and each ideal is a unique product of maximal ones. It is a remarkable fact that any simple CC-module is a sum of eigenspaces of the element YXYX (the set of eigenvalues for YXYX is given explicitly for each simple CC-module). This fact is crucial in finding the group AutFq(C) of FqFq-algebra automorphisms of CC and in proving that any two distinct Carlitz rings are not isomorphic (Cν≄CμCν≄Cμ if ν≠μν≠μ). The centre of CC is found explicitly, it is a UFD that contains countably many elements.