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I'm just learning basic ring theory and had a question about the definition of a principal ideal. For a commutative ring $R$ with unity, Fraleigh defines the principal ideal generated by $a\in R$ as $\langle a\rangle = \{ra:r\in R\}$. My question is, why do we need to assume that $R$ has a unity? It seems like $\langle a\rangle$ fits the criterion for an ideal without it. Is this the standard definition? If so, do we include the unity condition so that we can say that $a\in \langle a\rangle$? Also, if we require that every ring have a unity, wouldn't we have that the entire ring is a principal ideal of itself, since $R=\langle1\rangle$?

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    $\begingroup$ You can also talk about principal ideals in rings without unit. Nothing prevents you from doing that. $\endgroup$ Commented Apr 13, 2014 at 5:50
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    $\begingroup$ If you like to write ideals with angly-brackets, use \langle and \rangle, not < and >, so that you get $\langle a\rangle$ instead of the immensly uglier $<a>$ :-) $\endgroup$ Commented Apr 13, 2014 at 5:51
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    $\begingroup$ I am not familiar with non-unital ring theory, but the fact that $a\in Ra$ requires $1\in R$ seems relevant. There is another definition of principal ideal as that generated by a single element, where we define the ideal generated by a set to be the smallest ideal containing the set (so it is the intersection of all such). On this definition, $Ra$ would be different from $\langle a\rangle$ (the smallest ideal containing $a$) if $R$ doesn't have a $1$. In $R=2\Bbb Z$ for instance, $2R=4\Bbb Z$ but $\langle2\rangle=2\Bbb Z$. Also yes, $R=\langle1\rangle$ in any unital ring (why do you ask?). $\endgroup$ Commented Apr 13, 2014 at 5:57
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    $\begingroup$ «Defining things in the most general way» is a terribly bad way to write a textbook on basic ring theory (or basic anything, really)! And why would an author write a book and not restrict focus to whatevery it is he wants to talk about? It is his book! Your complaints about the book are quite strange! :-) $\endgroup$ Commented Apr 13, 2014 at 6:15
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    $\begingroup$ The purpose of a textbook on basic algebra is not to do everything, but to introduce the reader to, well, basic algebra. There are other books with other purposes. $\endgroup$ Commented Apr 13, 2014 at 6:21

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Getting this out of unanswered-limbo

why do we need to assume that R has a unity? It seems like ⟨𝑎⟩ fits the criterion for an ideal without it.

If $R$ does not have identity, then you're correct: $\{aR\mid r\in R\}$ is an ideal. However, one doesn't know whether or not $a\in aR$ when there's no identity in $R$. That's something we would really like $\langle a\rangle$ to satisfy.

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    $\begingroup$ Just saying that I like that you work on these. Ok in most cases the OP won't read the answer BUT since the questions will pop up in search results it is definitely worth it. $\endgroup$ Commented Oct 27, 2023 at 20:23

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