Definition:Generator of Module
Let $R$ be a ring.
Let $M$ be an $R$-module.
Let $S \subseteq M$ be a subset.
Some sources refer to a generator for rather than generator of. The two terms mean the same.
It can also be said that $S$ generates $M$ (over $R$).
Other terms for $S$ are:
- A generating set of $M$ (over $R$)
- A generating system of $M$ (over $R$)
Some sources refer to such an $S$ as a set of generators of $M$ over $R$ but this terminology is misleading, as it can be interpreted to mean that each of the elements of $S$ is itself a generator of $M$ independently of the other elements.