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For questions on modules over rings.
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Lifting a direct summand of a free module
given split short exact sequences
\begin{align*}
R \to S \times T &\to R^{n-1} \\
R/I \to S_I \times T_I &\to (R/I)^{n-1}
\end{align*}
the first inducing the second on quotients, consisting of free $R$-modules … ($R/I$-modules) $S= R^{n-1}$ and $T= R$ (resp. …