Mal'cev products of rings satisfying $x^n\approx x$ and idempotent semirings

Let $\text{Rn}$ be the variety of rings satisfying $x^n\text{eqid}x$, where $n\ge 2$, and let $\text{W}$ be a variety of idempotent semirings with commutative addition. The equality $\text{Rn}\circ \text{W}=\text{Rn}\vee \text{W}$ holds, and this variety has subvariety lattice $L(\text{Rn})\times L(\text{W})$. Its subdirectly irreducible members that are not additively idempotent have exactly one nontrivial ring component, which is a finite field. Such a member is determined by this field and an idempotent semiring with a multiplicative identity; a separation condition on unary polynomial functions of the latter characterizes subdirect irreducibility. Every subvariety has a finite identity basis. A four-element semiring generates a variety with subdirectly irreducible members of unbounded cardinality. The results extend the Mal’cev product theorem of Wang and Shao from the absorption subvariety to arbitrary $\text{W}\le \text{Slp}$.