Injectivity for cancellative semimodules
Let \(S=(S,+,\cdot)\) be a semiring with an absorbing zero \(0_S\) and an identity \(1\neq 0_S\). All \(S\)-semimodules \(A=(_SA,+)\) are assumed to be left-sided, unitary and zero-preserving, i.e. \(s0_A=0_A=0_Sa\) holds for all \(s\in S\) and \(a\in A\). Replacing in the definition of an injective \(R\)-module over a ring \(R\) ``ring'' by ``semiring'' and ``module'' by ``semimodule'', one obtains the usual concept of an injective \(S\)-semimodule. It has the disadvantage that for various well behaved semirings \(S\) the only \(S\)-injective \(S\)-semimodule is \(I=\{0\}\). For this reason the authors restrict all considerations to the category \({\mathfrak C}_S\) of all cancellative \(S\)-semimodules over a fixed semi-ring \(S\) (where \((S,+)\) may be cancellative or not) and obtain the following results for the resulting concept of (cancellative) \({\mathfrak C}_S\)-injective \(S\)-semimodules. Each \({\mathfrak C}_S\)-injective \(S\)-semimodule is an \(S\)-module. Each cancellative \(S\)-semimodule \(A\) is an \(S\)-subsemimodule of a \({\mathfrak C}_S\)-injective \(S\)-module, and \(A\) itself is \({\mathfrak C}_S\)-injective iff it is not large in any proper cancellative extension \(S\)-semimodule. Moreover, \(A\) has a \({\mathfrak C}_S\)-injective hull. As noted by the authors, cancellative \(S\)-semimodules are rather close to \(R\)-modules, and it remains open whether similar results can be obtained for other categories of \(S\)-semimodules.
scientific article; zbMATH DE number 1022786