Module
\((R, M, \cdot)\): left \(R\)-module.
\(R\): ring.
\(M\): group.
\(\cdot\): function from \(R \times M\) to \(M\) written as \(r \cdot m = rm\).
\(m, m_1, m_2\): elements of \(M\).
\(r, r_1, r_2\): elements of \(R\).
\(1m = m\): identity of \(R\) maps
\(r(m_1 + m_2) = rm_1 + rm_2\): operations in \(M\) align with operations in \(R\).
\((r_1 + r_2)m = r_1m + r_2m\): operations in \(R\) align with operations in \(M\).
\(r_1(r_2m) = (r_1 r_2)m\): operations carry over.
Right \(R\)-module: \(\cdot\) \(M \times R\) to \(M\).
Vector spaces
\(R\): a field \(F\).