On semimodules over commutative, additively idempotent semirings (Q5953366)

Let \((S,+,\cdot)\) be a commutative and additively idempotent semiring with absorbing zero and identity. A (right) \(S\)-semimodule \(P_S\) is projective if for any \(S\)-semimodules \(M_S\) and \(N_S\), any surjective \(S\)-homomorphism \(\varphi\colon M_S\to N_S\) and any \(S\)-homomorphism \(\psi\colon P_S\to N_S\), there exists an \(S\)-homomorphism \(\nu\colon P_S\to M_S\) such that \(\varphi\nu=\psi\). A surjective \(S\)-homomorphism \(\varphi\colon B_S\to A_S\) of \(S\)-semimodules is pure if for any finitely generated \(S\)-semimodule \(C_S\) and any \(S\)-homomorphism \(\eta\colon C_S\to A_S\) there exists an \(S\)-homomorphism \(\nu\colon C_S\to B_S\) such that \(\varphi\nu=\eta\). An \(S\)-semimodule \(A_S\) is called strongly flat if there exists a free \(S\)-semimodule \(F_S\) and a pure \(S\)-homomorphism \(\varphi\colon F_S\to A_S\). The main result generalizes a well known fact for bounded distributive lattices: For \((S,+,\cdot)\) as above the following conditions are equivalent. (1) All projective \(S\)-semimodules are free. (2) All strongly flat \(S\)-semimodules are projective. (3) \(S=\{0\}\).