Definition:Generator of Module/Unitary

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Definition

Let $R$ be a ring with unity.

Let $M$ be a unitary $R$-module.

Let $S \subseteq M$ be a subset.


$S$ is a generator of $M$ if and only if every element of $M$ is a linear combination of elements of $S$.


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:

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.


Also see


Sources