Module axioms

The module axioms define a (left) module $M$ over a unital ring $R$ as follows. One requires:

1. $(M,+)$ is an abelian group (so $+$ is a binary operation with associativity, commutativity, identity $0$, and inverses).
2. A scalar multiplication map $R\times M\to M$, $(r,m)\mapsto rm$, satisfying for all $r,s\in R$ and $m,n\in M$:
   • $(r+s)m = rm + sm$,
   • $r(m+n) = rm + rn$,
   • $(rs)m = r(sm)$,
   • $1_R m = m$.

These axioms encode “linearity” over a ring ; replacing $R$ by a field yields the vector space axioms .