A note on injectivity of \(K\)-groups (Q1176013)

This is a new attempt to introduce the concept of injectivity in terms of \(K\)-maps of \(K\)-sets and \(K\)-groups, where \(K\) is a near-ring. Since the terminology of the paper differs from the standard terminology in certain points, we need a series of definitions in the first place. \(K\) is assumed to be a right near-ring with identity 1. A nonempty set \(M\) is called \(K\)-set iff there exists a mapping \(M\times K\to M: (m,k)\mapsto mk\) such that (a) \((mk_ 1)k_ 2=m(k_ 1 k_ 2)\), (b) \(m1=m\) for all \(k_ 1,k_ 2\in K\), \(m\in M\). A \(K\)-set \(M\) is called a \(K\)-group iff \((M,+)\) is a group and \((m_ 1+m_ 2)k=m_ 1 k+m_ 2 k\), \(m0=0\) for all \(m,m_ 1,m_ 2\in M\), \(k\in K\). Note: The authors do not assume \(m(k_ 1+k_ 2)=mk_ 1+mk_ 2\). Consequently, only the multiplicative monoid of \(K\) is involved; the addition in \(K\) is inessential. --- Let \(M\) and \(M'\) be \(K\)-sets. A map \(f: M\to M'\) is called a \(K\)-map iff \(f(mk)=f(m)k\) for all \(m\in M\) and \(k\in K\). A \(K\)-homomorphism between \(K\)-groups is a homomorphism which is a \(K\)-map. Now let \(M\) be a \(K\)- group. Then \(M\) is called injective iff the following condition holds: Let \(A\), \(B\) be \(K\)-sets, \(\lambda: A\to B\) an injective \(K\)-map and \(g: A\to M\) a \(K\)-map, then there exists a \(K\)-map \(h: B\to M\) such that \(h\circ\lambda=g\). The authors give a characterization of injective \(K\)- groups analogous to Baer's characterization of injective modules, and they construct some type of injective hull of an Abelian \(K\)-group.