The Carlitz Algebras

The Carlitz $\mathbb{F}_q$-algebra $C=C_\nu$, $\nu \in \mathbb{N}$, is generated by an algebraically closed field $\CK$ (which contains a non-discrete locally compact field of positive characteristic $p>0$, i.e. $K\simeq \mathbb{F}_q[[ x,x^{-1}]]$, $q=p^\nu$), by the (power of the) {\em Frobenius} map $X=X_\nu :f\mapsto f^q$, and by the {\em Carlitz derivative} $Y=Y_\nu$. It is proved that the Krull and global dimensions of $C$ are 2, a classification of simple $C$-modules and ideals are given, there are only {\em 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 $\Aut_{\Fq}(C)$ of $\Fq$-algebra automorphisms of $C$ and in proving that two distinct Carlitz rings are not isomorphic $(C_\nu \not\simeq C_\mu$ if $\nu \neq \mu$). The centre of $C$ is found explicitly, it is a UFD that contains {\em countably many} elements.