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 $R=\langle1\rangle$?
• 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 $<a>$ :-) – Mariano Suárez-Álvarez Apr 13 '14 at 5:51
• 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?). – anon Apr 13 '14 at 5:57