Let $R$ be a one-dimensional locally analytically irreducible Noetherian domain with finite residue fields. In this note it is shown that if $I$ is a finitely generated ideal of the ring $\Int(R)$ of integer-valued polynomials such that for each $x \in R$ the ideal $I(x) =\{f(x) \mid f \in I\}$ is strongly $n$-generated, $n \geq 2$, then $I$ is $n$-generated, and some variations of this result.

We consider the rings of invariants $R^G$, where $R$ is the symmetric algebra of a tensor product between two vector spaces over the field $F_p$ and $G=U_n\times U_m$. A polynomial algebra is constructed and these invariants provide Chern classes for the modular cohomology of $U_{n+m}$.