Representation of near-ring Morita contexts and recognizing Morita near-rings (Q1375726)

Morita contexts for near-rings were defined by the present authors. A near-ring Morita context is a quadruple \(\Gamma=(\Gamma_{11},\Gamma_{12},\Gamma_{21},\Gamma_{22})\), such that for all \(i,j,k\in\{1,2\}\), there exists a multiplication \(\Gamma_{ik}\times\Gamma_{kj}\to\Gamma_{ij}\) which satisfies the natural notions of associativity and right distributivity. If \(\Gamma=(\Gamma_{11},\Gamma_{12},\Gamma_{21},\Gamma_{22})\) and \(\Gamma'=(\Gamma_{11}',\Gamma_{12}',\Gamma_{21}',\Gamma_{22}')\) are two Morita contexts, a Morita context homomorphism is a quadruple \(\alpha=(\alpha_{11},\alpha_{12},\alpha_{21},\alpha_{22})\), where \(\alpha_{ij}\colon\Gamma_{ij}\to\Gamma_{ij}'\) is a group homomorphism such that \(\alpha_{kj}(xy)=\alpha_{ki}(x)\alpha_{ij}(y)\) for all \(x\in\Gamma_{ki}\), \(y\in\Gamma_{ij}\). Embeddings and isomorphisms are defined in the obvious way. If \(\Gamma\) is a Morita context, the Morita context near-ring \(M_2(\Gamma)\) is defined in terms of mappings of the group \(\left[\begin{smallmatrix}\Gamma_{11} &\Gamma_{12}\\ \Gamma_{21}&\Gamma_{22}\end{smallmatrix}\right]\) into itself. If \(\Gamma=(L,G,H,R)\) is a Morita context, we define \(M_R(G)=\{f\colon G\to G\mid f(gr)=f(g)r\;\forall g\in G,\;r\in R\}\) and \(M_R(G,R)=\{f\colon G\to R\mid f(gr)=f(g)r\;\forall g\in G,\;r\in R\}\). Then \(\Gamma^\#=(M_R(G),G,M_R(G,R),R)\) is a Morita context. It is shown that there exists a homomorphism \(\alpha\) of \(\Gamma\) into \(\Gamma^\#\), and that \(\alpha\) is an embedding if and only if \((0:G)_L=0\) and \((0:G)_H=0\). Necessary and sufficient conditions are given for \(\alpha\) to be an isomorphism. If \(A\) is a near-ring with unity, and \(e\) is an idempotent of \(A\), let \(e_1=e\) and \(e_2=1-e\). For \(i=1,2\), let \(D_i=\{e_1ae_i+ebe_i\mid a,b\in A\}\) and let \(S\) be the subnear-ring of \(A\) generated by \(\{e_iae_j\mid 1\leq i,j\leq 2\}\). It is shown that \(A\) is isomorphic to some Morita context near-ring \(M_2(\Gamma)\) where \(\Gamma=(\Gamma_{11},\Gamma_{12},\Gamma_{21},\Gamma_{22})\), \(\Gamma_{ii}\) are near-rings with unity and all modules are unital if and only if: (i) \(ea+(1-e)b=(1-e)b+ea\) for all \(a,b\in A\); (ii) \((0:D_1)_A\cap(0:D_2)_A=0\); and (iii) \(S=A\).