Definition talk:Generated Ideal of Ring

I found a definition and proposition of generated ideal and principal ideal.


 * : Chapter $6$. Rings: $\S 2$. Homomorphisms and Ideals

Although the set $I$ of linear combinations of elements of $S$ is an ideal of $R$, but it is not the definition of generated ideal in this book. In addition, if $R$ is without unity, it is possible to be $S \nsubseteq I$. What do you think of ideal generation? --Fake Proof (talk) 05:46, 25 July 2021 (UTC)


 * Beware, this page was edited into its current form by an omniscient idiot. There are three different definitions here, but they are not equivalent because they do not apply to the same objects.


 * Feel free to analyse the differences between what you see on this page (or rather, suite of pages) and what you have in your source work, and see whether you can reconcile those differences. --prime mover (talk) 06:52, 25 July 2021 (UTC)


 * I agree with Definition 1, but I want to separate it into two equivalent ones. The definitions of generated ideal and principal ideal in the book (Definition $6.2.28$) are:


 * The smallest ideal containing a subset $S$ is called the ideal generated by $S$. The smallest ideal containing a single element $x \in R$ is called the principal ideal generated by $x$.


 * There is a theorem (Proposition $6.2.27$) for generated ideals.


 * Let $\gen S$ denote the additive subgroup of $R$ generated by $S$. Then $\gen S + R S + S R + R S R$ is the smallest ideal of $R$ containing $S$. (Definitions of $R S, S R, R S R$ are in the book.)


 * Instead of definitions 2 and 3, it can be another equivalent definition, or a theorem.


 * Then some theorems such as Principal Ideal is Ideal and Principal Ideal is Smallest Ideal seem to be meaningless with the new definition of principal ideal. --Fake Proof (talk) 08:08, 25 July 2021 (UTC)