  Let ${\bf Sr}(n, 1)$ denote the ai-semiring variety defined by the identity
$x^n\approx x$, where $n>1$. We characterize all subdirectly irreducible
members of a semisimple subvariety of ${\bf Sr}(n, 1)$. Based on this result,
we prove that ${\bf Sr}(n, 1)$ is hereditarily finitely based (resp.,
hereditarily finitely generated) if and only if $n<4$ and that the lattice of
subvarieties of ${\bf Sr}(n, 1)$ is countable if and only if $n<4$. Also, we
show that the class of all locally finite members of ${\bf Sr}(n, 1)$ forms a
variety and so we affirmatively answer the restricted Burnside problem for
${\bf Sr}(n, 1)$. In addition, we provide a simplified proof of the main result
obtained by Gajdo\v{s} and Ku\v{r}il (Semigroup Forum 80: 92--104, 2010).
