Goldie's theorem is a theorem in noncommutative ring theory that gives a clear picture of semiprime Noetherian rings (actually a slightly broader class). Let $R$ be a semiprime Noetherian ring. The theorem shows that the set of nonzerodivisors of $R$ can all be inverted, and after you invert them you get a semisimple Artinian ring.
I'm curious to what extent an analogous result holds for semigroups. The only relevant paper I found was a paper from Semigroup Forum, Goldie's Theorem for Semigroups, by Song Guangtian. Despite the title, the result seems to offer room for improvement. The analogue for simple Artinian rings in the theorem is Rees matrix semigroups (with zero), but the result requires a fussy additional condition (condition 3 in Theorem 4.11). Maybe a sharper analogue can be formulated by considering a wider class of semigroups than Rees matrix semigroups?
