Definition talk:Ring (Abstract Algebra)

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It's a trivial point, but my impression was that usually $R^*=$ the group of units of a ring, which usually gets defined by $R-0$ in texts when working only with fields. Could just be me though Linus44 08:38, 12 February 2011 (CST)

Recommend that when we need to use the concept of either the set of units or the ring less unity, if we use $R^*$ we make sure we define it first thing on the page. "Let $R$ be a ring and $R^*$ the set of units of that ring" or whatever (neater of course and linking to the appropriate definitions ;-) and then we'll know where we stand. --prime mover 14:26, 12 February 2011 (CST)
... I've put a note in on this page to indicate that there are different interpretations. --prime mover 14:30, 12 February 2011 (CST)

Isn't the additive group $(R,∗)$ supposed to be Abelian? --Alec (talk) 14:09, 30 May 2011 (CDT)

't Appears as were the reference lost under refactoring operations. A section 'Also defined as' would be good, I think. --Lord_Farin 18:39, 1 June 2012 (EDT)
What reference lost? Can't find it. --prime mover 18:54, 1 June 2012 (EDT)
Sure thing. But as this can be deduced from the other axioms, I didn't include it as an axiom but added it as a theorem. This has now been referred to on the page so as to allay confusion. --prime mover 14:27, 30 May 2011 (CDT); I couldn't find the reference on the page. --Lord_Farin 03:25, 2 June 2012 (EDT)
That's because some stupid old man had removed it when he was having one of his feebleminded episodes. It has been restored and improved. --prime mover 04:07, 2 June 2012 (EDT)

Semigroup under $\circ$?

According to Encyclopedia of Math, $(R, \circ)$ need only be a "groupoid" (which I take to mean magma), and in particular $\circ$ needn't be associative. --Dfeuer (talk) 07:17, 9 January 2013 (UTC)

Nightmare. Okay, I'm on the case. Moving all instances of "Ring (Abstract Algebra)" to "Associative Ring". I'll leave it up to someone else to amend all the pages that are affected. Dfeuer, you noticed it, I'll leave it up to you to do the latter task. --prime mover (talk) 07:31, 9 January 2013 (UTC)
It's past my bedtime, but I'll see what I can do tomorrowish. --Dfeuer (talk) 07:43, 9 January 2013 (UTC)
Why?! Fk, it could have been as simple as amending this page with an "Also defined as" and creating a page for "Non-associative Ring". Almost all mathematicians (in particular, all the ones I've met) impose that a ring be associative. We might risk losing audience here if we don't link everything up properly. --Lord_Farin (talk) 08:48, 9 January 2013 (UTC)
Sorry, I thought this was what we needed to do. I admit the Encyclopedia of Math is the only place where it is suggested that the product need not be associative. I just assumed it was a recent development in abstract algebra. --prime mover (talk) 09:05, 9 January 2013 (UTC)
Admittedly such basic algebra is not high on my list of interest (if it appears at all) and I rather deal with Boolean or operator algebras (which have some more structure than a mere ring). It could be that it's a recent development. It's just an insane amount of work to do because of one (web) source. I'd rather have more than one. --Lord_Farin (talk) 09:10, 9 January 2013 (UTC)
I have no idea about algebra. I just saw that, and figured we should probably mention it as an alternative definition. When you freaked out, I figured the situation must be worse than I had realized. I don't own recent algebra sources (and the only one I can find at the moment is M&S, which defines a ring as a ring with unity), so I can't help with the literature check. --Dfeuer (talk) 09:12, 9 January 2013 (UTC)
Cursory search through my ebook library revealed that Vinberg in his 2003 "A Course in Algebra" indeed introduces a ring without associativity. 't Be noted that I sifted to several other recent texts that did impose associativity. If you're up for it, go ahead. I'd rather not, but then I won't be involving myself in this matter. Your party. --Lord_Farin (talk) 09:26, 9 January 2013 (UTC)

Lord_Farin, algebra isn't really my cup of tea either. I think it's important to offer a definition of a ring without associativity, but I don't much care how "ring" is defined. --Dfeuer (talk) 09:28, 9 January 2013 (UTC)

Tell me what you want me to implement then, and I'll get on with it. I can't do anything unless you tell me exactly how you want it to be written. --prime mover (talk) 09:50, 9 January 2013 (UTC)
Huh??? This started with me mentioning that a certain source defines it differently, and suggesting we should have that, and moves on to you getting snarky. No need. I'll be happy to do it myself, but not now. --Dfeuer (talk) 09:58, 9 January 2013 (UTC)
But you, prime_mover, do have to decide what things will be called. Do you want "ring/associative ring", "ring/non-associative ring", or something else? --Dfeuer (talk) 10:00, 9 January 2013 (UTC)
Second suggestion is strongly favoured by me. I think it scares people (e.g. those coming from applied directions) less. --Lord_Farin (talk) 10:11, 9 January 2013 (UTC)
The biggest advantage of that approach is that it doesn't make a bunch of pages wrong all of a sudden. Once we create "non-associative ring" (or whatever we call it), we can change the relevant pages at our leisure. --Dfeuer (talk) 10:48, 9 January 2013 (UTC)
I agree that rings are/should be associative. Anything else would just lead to lots of confused users changing it back to the normal definition. A simple addition to the "Also defined as" section should suffice. --Alec (talk) 17:37, 10 January 2013 (UTC)