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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 irreduciblemembers 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 ofsubvarieties of ${\bf Sr}(n, 1)$ is countable if and only if $n<4$. Also, weshow that the class of all locally finite members of ${\bf Sr}(n, 1)$ forms avariety and so we affirmatively answer the restricted Burnside problem for${\bf Sr}(n, 1)$. In addition, we provide a simplified proof of the main resultobtained by Gajdo\v{s} and Ku\v{r}il (Semigroup Forum 80: 92--104, 2010).

The Burnside ai-semiring variety defined by $x^n\approx x$