Solving linear systems appears hard in semirings.

In $\mathbb{N}_0 (+,\times)$ it is NP-complete via reduction to 1-in-3 SAT.

In the min-plus semiring the complexity is $ NP \cap coNP$ according to this

Are there nontrivial semirings where solving linear systems is in P?

Is they exist I would be interested what constraints they can encode.

**Added clarification**

By "linear system" mean a system of linear equations of the form: $$ c_1 \otimes x_1 \oplus c_2 \otimes x_2 \oplus \cdots \oplus c_n \otimes x_n = c_{n+1}$$ where $c_i$ are constants and $x_i$ are variables. If the standard definition is other, say $A x = B x$, would accept an answer about the standard definition.

By "solving a linear system" system mean finding at least one solution or claim that no solution exist. Though finding some kind of "basis" for all solutions would be interesting too.