Goldie's theorem for semigroups (Q687637)

Let \(S\) be a semigroup with 0. A subsemigroup \(S \subseteq Q\) is a called a left order of \(Q\) and \(Q\) is called a classical left quotient semigroup of \(S\) if every cancellable element in \(S\) is a unit of \(Q\) and for each \(q \in Q\) there exist \(a,b \in S\) such that \(a\) is cancellable in \(S\) and \(q = a^{-1}b\). In analogy to the Artin-Wedderburn-Theorem stating that simple artinian rings are matrix rings over a division ring, the author considers row monomial matrix monoids \(M_ n(G^ 0)\) over \(G^ 0\), a group \(G\) with zero 0. It is well known that they are isomorphic to the endomorphism monoid of \(n\)-dimensional free \(G^ 0\)-systems (\(G^ 0\)- acts). An \(S\)-system \(M\) is called finite dimensional if it cannot be decomposed into an infinite coproduct of subsystems. Now the number of components in the finest possible decomposition of \(M\) is called the Goldie dimension \(\dim_ S(M)\) of \(M\). A semigroup \(S\) is called left Goldie if it is as a left \(S\)-system of finite Goldie dimension (which is trivial if \(S\) is a monoid) and satisfies acc on left and on right equalizer congruences, where \(\rho \subseteq S\times S\) is a left equalizer congruence if for some left \(S\)-system \(M\) one has \((a,b) \in \rho\) iff \(am = bm\) for all \(m\in M\). The main theorem characterizes semigroups \(S\) which have the classical left quotient semigroup \(Q = M_ N(G^ 0)\) by three conditions one of which being \(S\) is an \(n\)-dimensional left Goldie semigroup.