Definition:Generator of Module/Also known as
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Generator of Module: Also known as
A generator of a module is also known as a spanning set.
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.