We study ultraproducts of finite residue rings \(\prod \nolimits _{n\in {\mathbb {N}}} {\mathbb {Z}}/n{\mathbb {Z}}\diagup {\mathcal {U}} \) where \({\mathcal {U}}\) is a non-principal ultrafilter. We find sufficient conditions of the ultrafilter \({\mathcal {U}}\) to determine if the resulting ultraproduct \(\prod \nolimits _{n\in {\mathbb {N}}} {\mathbb {Z}}/n{\mathbb {Z}}\diagup {\mathcal {U}}\) has simple, NIP, \(\mathrm {NTP}_{2}\) but not simple nor NIP, or \(\mathrm {TP}_{2}\) theory, noting that all these four cases occur.

我们研究有限剩余环的超积\（\ prod \ nolimits _ {n \ in {\ mathbb {N}}} {\ mathbb {Z}} / n {\ mathbb {Z}} \ diagup {\ mathcal {U} } \）其中\（{\ mathcal {U}} \）是非主要的超滤镜。我们找到超滤子\（{\ mathcal {U}} \）的充分条件，以确定所得的超产品\（\ prod \ nolimits _ {n \ in {\ mathbb {N}}} {\ mathbb {Z}} / n {\ mathbb {Z}} \ diagup {\ mathcal {U}} \）具有简单，NIP，\（\ mathrm {NTP} _ {2} \），但不是简单，NIP或\（\ mathrm { TP} _ {2} \）理论，请注意这四种情况均会发生。