Number of operations to compute product of pairwise sums over a commutative semiring
Not an answer to your exact question, but may be useful if you happen to have a semiring with cancellation: if you have an algorithm which works in any ring, you can apply that algorithm to the difference ring $S-S$ consisting of subtraction pairs $s - t$.  The result will have the form $s - t$ where $s,t \in S$, and the answer will be $d \in S$ s.t. $s = d + t$.  Thus, in addition to semiring operations, you'd need to perform a single cancellation step at the end.