Definition talk:Principal Ideal of Ring

As multiplication is not necessarily associative, this will only be a left ideal. The proof of Principal Ideal is an Ideal skims over this. I suppose it is an omission in the definition. --Lord_Farin 14:38, 23 October 2011 (CDT)
 * Yes, you're right - I've seen a distinction being made between left & right ideals in various sites on the web, but not in any of the books I've actually come into contact with. Feel free to attack this area with your usual gusto. --prime mover 14:41, 23 October 2011 (CDT)

My approach is radically different:

Proposed definition: A principal ideal is an ideal that is generated by a single element $a$.

This is (AFAIK) the usual definition of a principal ideal. --Lord_Farin 15:15, 23 October 2011 (CDT)


 * All I've got is what my books tell me and they clearly give an incomplete definition. Your radically different approach is not so different from the definition given, if you click on "generated" and it goes to "Generator of an Ideal" etc. Except that the definition of ideal that you have is the union of the left and right ideal, it (shrug) looks the same to me. What's your source work? --prime mover 15:45, 24 October 2011 (CDT)


 * For almost all parts of mathematics I have come to know, I have learned from lecture notes used on Utrecht University. No different for rings. I haven't found an online version however. --Lord_Farin 08:29, 25 October 2011 (CDT)


 * Lecture notes are good. Haven't determined a template for their citation in the Sources section yet (I believe they probably need a presentation somewhat different from a book or a website. No worries. My own formal studies never went down the route of abstract algebra any further than groups, and everything I know further than that has been self-taught. So I'll let you consider whatever restructure you think necessary.
 * Utrecht: never been. But the company I work for has a contract in Woerden at the moment, I may be offered the opportunity of going. Hope so, I'm fond of Nederland. --prime mover 12:58, 25 October 2011 (CDT)

I think I understand what's going on. In the initial work I consulted to put this page together (Whitelaw), a principal ideal is defined only for a commutative ring with unity. Then I made the mistake of consulting Warner, which does not require the ring to be commutative, but still uses the same conditions - as you say, this appears to be wrong. I will take another look at this later. --prime mover 01:51, 28 October 2011 (CDT)


 * Note that I corrected the definition as to make the thing a subgroup in the first place. --Lord_Farin 04:02, 28 October 2011 (CDT)


 * Quick question: do we need $r^i \circ a \circ s^i$ or can we just get away with $r \circ a \circ s$? $r^i$ and $s^i$ are in $R$ anyway, so the indices surely are not needed? And again, I don't see why we need the sum, because it all just boils down to (some element of $R$) $\circ a \circ$ (another element of $R$) and hence $r \circ a \circ s$ (for all $r, s \in R$) should be sufficient. Or am I missing something? --prime mover 17:45, 28 October 2011 (CDT)


 * Yes, for in general $r \circ a \circ s + s \circ a \circ r$ may not be expressible as some $r'\circ a \circ s'$. It is tough to imagine such a ring though; I will try to think of an example. --Lord_Farin 18:21, 28 October 2011 (CDT)