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We establish an explicit formula for the number $C_n(q)$ of ideals ofcodimension $n$ of the algebra ${\mathbb F}_q[x,y,x^{-1}, y^{-1}]$ of Laurentpolynomials in two variables over a finite field of cardinality $q$. Thisnumber is a palindromic polynomial of degree $2n$ in $q$. Moreover, $C_n(q) =(q-1)^2 P_n(q)$, where $P_n(q)$ is another palindromic polynomial; the latteris a $q$-analogue of the sum of divisors of $n$, which happens to be the numberof subgroups of ${\mathbb Z}^2$ of index $n$.

Counting the ideals of given codimension of the algebra of Laurent polynomials in two variables