Injective hulls of semimodules over additively-idempotent semirings (Q1322588)

A semiring \((R,+,\cdot)\) is a nonempty set \(R\) where \((R,+)\) is a commutative monoid with additive identity 0, \((R,\cdot)\) is a monoid with multiplicative identity 1, \(0r = 0 = r0\) for all \(r\in R\) and multiplication distributes over addition from either side. A semiring \(R\) is said to be additively-idempotent, if \(r + r = r\) for all \(r \in R\). A commutative monoid \((M,+,0_ M)\) is called a left semimodule over a semiring \(R\) (briefly left \(R\)-semimodule) if there is a scalar multiplication \((r,m) \to rm\) from \(R \times M\) to \(M\) satisfying standard conditions. A left \(R\)-semimodule \(I\) is said to be injective if, given a subsemimodule \(N\) of a semimodule \(M\), any \(R\)-homomorphism from \(N\) to \(I\) can be extended to an \(R\)-homomorphism from \(M\) to \(I\). It is well known that every left module over a ring \(R\) can be embedded into an injective left \(R\)-module. In the present paper the analogous result is obtained for a left semimodule over an additively-idempotent semiring.