On ideal class groups of totally degenerate number rings

Let χ(x)∈Z[x] be a monic polynomial whose roots are distinct integers. We study the ideal class monoid and the ideal class group of the ring Z[x]/(χ(x)). We obtain formulas for the orders of these objects, and study their asymptotic behavior as the discriminant of χ(x) tends to infinity, in analogy with the Brauer-Siegel theorem. Finally, we describe the structure of the ideal class group when the degree of χ(x) is 2 or 3.