The Carlitz algebras

The Carlitz image-algebra C=Cν, image, is generated over an algebraically closed field image (which contains a non-discrete locally compact field of positive characteristic p>0, i.e. image, q=pν), by the (power of the) Frobenius map X=Xν:fmaps tofq, and by the Carlitz derivative Y=Yν. 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 image of image-algebra automorphisms of C and in proving that any two distinct Carlitz rings are not isomorphic (Cνnot asymptotically equal toCμ if ν≠μ). The centre of C is found explicitly, it is a UFD that contains countably many elements.