Remark 115.4.4 (Projective resolutions). Let $R$ be a ring. For any set $S$ we let $F(S)$ denote the free $R$-module on $S$. Then any left $R$-module has the following two step resolution

\[ F(M \times M) \oplus F(R \times M) \to F(M) \to M \to 0. \]

The first map is given by the rule

\[ [m_1, m_2] \oplus [r, m] \mapsto [m_1 + m_2] - [m_1] - [m_2] + [rm] - r[m]. \]

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