Defining principal ideals in commutative rings without identity

Question

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 $R=\langle1\rangle$?

Answer 1

Getting this out of unanswered-limbo

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why do we need to assume that R has a unity? It seems like ⟨⟩ 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.