内射T-模上的广义Schanuel引理

Generalized Schanuel Lemma on Injective T-Modules

摘要: 本研究桁架上的内射模，证明了若以下两个T-模序列正合$\begin{array}{ccc}★& \stackrel{}{\to }& M\\ & & \\ ★& \stackrel{}{\to }& M\end{array}\begin{array}{ccc}\stackrel{i}{\to }& E_1& \stackrel{f_1}{\to }\\ & & \\ \stackrel{i^{\prime }}{\to }& {E^{\prime }}_1& \stackrel{{f^{\prime }}_1}{\to }\end{array}\begin{array}{ccc}{E}_2& \stackrel{\pi }{\to }& Q\\ & & \\ {E^{\prime }}_2& \stackrel{{\pi }^{\prime }}{\to }& Q^{\prime }\end{array}\begin{array}{c}\stackrel{}{\to }\\ \\ \stackrel{}{\to }\end{array}\begin{array}{c}★\\ \\ ★\end{array}$其中$E_i,{E^{\prime }}_i$ 为内射T-模，且$Abs\left(E_i\right)\ne \varnothing ,Abs\left({E^{\prime }}_i\right)\ne \varnothing \left(i=1,2\right)$ ，则$Q^{\prime }\times E_2\times {E^{\prime }}_1\cong Q\times {E^{\prime }}_2\times E_1$ 。

Abstract: In this paper, we study injective modules over trusses. It is proven that if the following two sequences of T-modules are exact $\begin{array}{ccc}★& \stackrel{}{\to }& M\\ & & \\ ★& \stackrel{}{\to }& M\end{array}\begin{array}{ccc}\stackrel{i}{\to }& E_1& \stackrel{f_1}{\to }\\ & & \\ \stackrel{i^{\prime }}{\to }& {E^{\prime }}_1& \stackrel{{f^{\prime }}_1}{\to }\end{array}\begin{array}{ccc}{E}_2& \stackrel{\pi }{\to }& Q\\ & & \\ {E^{\prime }}_2& \stackrel{{\pi }^{\prime }}{\to }& Q^{\prime }\end{array}\begin{array}{c}\stackrel{}{\to }\\ \\ \stackrel{}{\to }\end{array}\begin{array}{c}★\\ \\ ★\end{array}$ where $E_i,{E^{\prime }}_i$ are injective T-modules with $Abs\left(E_i\right)\ne \varnothing ,Abs\left({E^{\prime }}_i\right)\ne \varnothing$ for $\left(i=1,2\right)$ , then $Q^{\prime }\times E_2\times {E^{\prime }}_1\cong Q\times {E^{\prime }}_2\times E_1$ .