Weighted automata are a structure that has many applications in computer science (speech recognition, natural language processing, image processing, quantitative modelling, etc). Weighted language equivalence is one possible semantics for these weighted automata. This equivalence is decidable for automata with weights on a field, but the proof relies heavily on the properties of fields. However, weighted automata can be defined more generally on a semiring, or even on structures where the addition is defined only partially. This talk explains how the result of decidability for weighted language equivalence for automata on fields can be extended to a decidability result for a certain class of semirings and ``partial semirings''.