Ultraproducts and ultra-limits of near-rings (Q1061823)

If N is the direct product of the near-rings \(N_{\alpha}\) (\(\alpha\in A)\) and if \({\mathfrak F}\) is a filter on A, we define an ideal I(\({\mathfrak F}):=\{(...,i_{\alpha},...)\in N|\) \(\{\) \(\alpha\) \(|\) \(i_{\alpha}=0\}\in {\mathfrak F}\}\). The factor N/I(\({\mathfrak F})\) is denoted by \(\prod N_{\alpha}/{\mathfrak F}\) and called the \({\mathfrak F}\)- filterproduct of the family \((N_{\alpha})_{\alpha \in A}\). If \({\mathfrak F}\) is a maximal filter \((=ultrafilter)\), \(\prod N_{\alpha}/{\mathfrak F}\) is called an ultraproduct. One big advantage of ultraproducts is the fact that elementary properties \((=sentences\) in a first-order language for near-rings) of the \(N_{\alpha}\) carry over to their ultraproduct; so ultraproducts of near-rings, near-fields, etc. are again of this type. This paper shows that some higher-order properties also extend from the \(N_{\alpha}\) to their ultraproduct. For instance, it is shown that if \(\Gamma_{\alpha}\) is a unitary \(N_{\alpha}\)-group for each \(\alpha\) then \(\prod N_{\alpha}/{\mathfrak F}\) is 2-primitive on \(\prod \Gamma_{\alpha}/{\mathfrak F}\) (defined analogously as above) if and only if \(\{\) \(\alpha\) \(|\) \(N_{\alpha}\) is 2-primitive on \(\Gamma_{\alpha}\}\in {\mathfrak F}\). The proof needs other results on ultraproducts of N-groups of type \(\nu\) and of \(\nu\)-modular left ideals. Also, it is shown that ultraproducts of distributively generated \((=:d.g.)\) near-rings \(N_{\alpha}\) are again d.g. if there is a fixed k such that each element of each \(N_{\alpha}\) is the sum of at most k (anti-) distributive elements (this is not true without this assumption on k). Finally, ultrapowers and ultralimits are studied. Briefly: ultralimits of 2-primitive near-rings are again 2-primitive.