Both my book and lecturer have in the definition a ring omitted the requirement of a unity.(adsbygoogle = window.adsbygoogle || []).push({});

I was reading in my book about ideals, more specifically principal ideals. I stumbled over a formula that differed by whether or not the ring had a unity. As an example I state the two for principal left ideals for a ring R:

[tex](a)_l = \{ar+na | r \in R, n \in \mathbf{Z} \}[/tex]

[tex](a)_l = \{ar | r \in R,\}[/tex]

Why is the extra term omitted if the ring does not have a unity? I bet the explanation is easy answer, but despite how hard I am looking at it, I cannot figure it out.

I also looked at an example. I took 2Z which has no unity and looked at 4Z which is a principal left (or right) ideal generated by 4. The formula then dictates that:

[tex](4)_l = \{ 4r + 4n | r \in 2 \mathbf{Z}, n \in \mathbf{Z} \}[/tex]

But then why not just say that:

[tex](4)_l = \{ 4n | n \in \mathbf{Z} \} = 4 \mathbf{Z}[/tex]

Am I doing something horrendously wrong here?

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# Principal ideals of rings without unity

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