Talk:Ring homomorphism

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[edit] f(1)=1

Correct me if I'm wrong, but the condition f(1)=1 on the ring homomorphism is wrong/misleading. A ring homomorphism is a group homomorphism between the additive subgroups of R and S s.t. it preserves multiplication; hence, f(0_R)=0_S, but f(1_R) does not necessarily map to 1_S. I'm not sure what definition of a ring homomorphism is being used here, but it's much stronger than the usual definition I've seen. 131.215.172.175 09:33, 10 January 2007 (UTC)

The rings in this article are assumed to have unity. It is very natural I would think that any ring homomorphism maps 1 to 1. Oleg Alexandrov (talk) 16:43, 10 January 2007 (UTC)
A ring homomorphism f: R -> S need not map 1_R to 1_S. Take R to be the integers, S to be 2 x 2 matricies with real entries, and the map f: x --> [[ x,0 ][ 0,0 ]] as an example. Mascdman 05:55, 12 January 2007 (UTC)
Yeah, I am aware of that. You could even simpler take f(r)=0 for all r. But I doubt such homomomrphisms are that important. In your example, for instance, you could restrict f to make it surjective, then its image would be a ring with unity and f would be a homomorphism mapping 1 to 1. Oleg Alexandrov (talk) 13:51, 12 January 2007 (UTC)
Why are all rings assumed to have unity? This doesn't sound standard to me. One would like to have 0 as a ring, and talk about the zero map, for example. Also, one would like to have the linear map \times n \colon \mathbb{Z} \to \mathbb{Z} to be a ring map, for all integers n, not just for n= 1! Actually, I just looked up the page on subrings, and I noted that n \mathbb{Z} is declared there not to be a subring of the integers!! This is getting weirder and weirder. In particular, this says ideals need not be subrings. Isn't this silly? Turgidson 17:21, 12 January 2007 (UTC)
See Wikipedia:WikiProject Mathematics/Conventions. Also, to me personally, it sounds very standard that rings must have unity. I never heard ideals be called "subrings". Oleg Alexandrov (talk) 17:23, 12 January 2007 (UTC)
OK, thanks for pointing out that convention. Sorry, I'm relatively new to Wikipedia, so I don't know in detail all the various mathematical conventions adopted at Wikipedia. Just to make sure: are these conventions set in stone, for all eternity, or are they still open to debate? I maintain that this particular convention is not standard, at least in the mathematical world. In fact, I'm teaching right now a course on Rings and Fields, and the (absolutely canonical) textbook takes what I believe to be the standard convention, namely, that rings need not have 1, and even if they do, ring homomorphisms need not respect 1. Now, I could give many more references to that effect (in case my argument above was not convincing enough), but just to give an existence proof that this convention I am talking about may well be adopted, even on wikipedia-related sites, see [1] and [2]. Turgidson 17:45, 12 January 2007 (UTC)
I don't know if the conventions are set in stone. The forum for discussing things is Wikipedia talk:WikiProject Mathematics. You could give it a try. Oleg Alexandrov (talk) 04:12, 13 January 2007 (UTC)
The condition f(1) = 1 is not natural, because it would exclude the trivial homomorphism f(x) = 0 that exists between any two rings. Also, many other interesting homomorphisms will be excluded, for example one that takes any 2x2 matrix, and turns it into a 3x3 matrix by filling with zeroes. Albmont 11:43, 5 April 2007 (UTC)
Turgidson and Mascdman have a point there re rings w/o a unit and non-unital homomorphisms. article oughtta be modified. Mct mht 08:15, 28 March 2007 (UTC)
My introductory abstract algebra textbook, Fraleigh's A First Course in Abstract Algebra, makes a distinction between "unital" and "non-unital" homomorphisms. By Fraleigh's definition, a "ring homomorphism" does not normally require f(1)=1. But if f(1)=1, then he calls it a "unital ring homomorphism". My lecturer doesn't care for the distinction and only talks about unital homomorphisms, but as long as we're being flexible about unity I think Fraleigh's approach on this is reasonable. It avoids artificially excluding interesting homomorphisms, while still letting us talk specifically about the f(1)=1 case when/if we want to. Crispy 11:26, 22 October 2007 (UTC)