Definition talk:Generator of Module

This page also needs some corrections, which I will try to do soon. --Anghel (talk) 20:58, 17 January 2023 (UTC)

If you are expecting to make material changes to the conditions of the definitions, these may need to be discussed. For a start, the declaration that the underlying ring may need to be a ring with unity is something which needs to be properly sorted out. We are in danger of losing precision. --prime mover (talk) 22:54, 17 January 2023 (UTC)

I will upload a counterexample that shows in a ring without unity, the submodule generated by $S$ may not be equal to the submodule consisting of all linear combinations of $S$. This should reveal why we need to condition for $R$ to be a ring with unity.

Second, I would like to change this definition so it says:

"Let $R$ be a ring.

Let $M$ be an $R$-module.

Let $H$ be a submodule of $M$.

Let $S \subseteq H$ be a subset.

$S$ is a generator of $H$ if and only if $H$ is the submodule generated by $S$."

...and so on for the remaining definitions, to make it explicit that the submodule generated may be a submodule of $M$, as in the definition of Definition:Generated Submodule. --Anghel (talk) 11:38, 18 January 2023 (UTC)