| definiendum | ⟨M,R,∗⟩∈leftModule(M,R) |
| context | M∈abelianGroup(M) |
| context | R∈ring(R) |
| context | ∗:R×M→M |
Now denote the addition in th group M by “+” as usual, and the addition and multiplication in the ring R by “ˆ+” and “ˆ∗”, respectively.
| postulate | r∗(x+y)=(r∗x)+(r∗y) |
| postulate | (r ˆ+ s)∗x=(r∗x)+(s∗x) |
| postulate | (r ˆ∗ s)∗x=r∗(s∗x) |
“∗” is an action of the ring on the group from the left. If the ring is commutative, then one need not distinguish between left- and right module.
One generally speaks of an R-left-module over M. Here R and M are just sets.