For questions on modules over rings.

Let $R$ be a ring. Then a left $R$-module is an abelian group $(M, +)$ together with an operation $R\times M \to M$, $(r, m) \mapsto r\cdot m$ such that

  • $r\cdot(m_1 + m_2) = r\cdot m_1 + r\cdot m_2$
  • $(r_1 + r_2)\cdot m = r_1\cdot m + r_2\cdot m$
  • $(r_1r_2)\cdot m = r_1\cdot(r_2\cdot m)$

If $R$ has an identity, $1_R$, then we require $1_R\cdot m = m$; this is sometimes called a unital left $R$-module.

There is a completely analogous definition for right $R$-modules.

If $R$ is a field, then $R$-modules correspond to vector spaces over $R$. If $R = \mathbb{Z}$, then $R$-modules correspond to abelian groups.