Wikipedia:Reference desk/Archives/Mathematics/2012 October 29

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October 29

Identify this ring

Is the ring $\mathbb {Z} [i]/(2+i)$ isomorphic to a more familiar ring? I have experimented and I see nothing the cosets have in common particularly. I don't think it's isomorphic to the trivial ring since, for instance, $i\in \mathbb {Z} [i]$ but $i\notin (2+i)$, nor is it isomorphic to $\mathbb {Q}$ which is what you would get if all elements in the same coset lied on a line which passes though the origin on an argand diagram (which is plausible at first) since for instance $5$ is in the same coset as $(2+i)$ $5=(2-i)(2+i)$. — Preceding unsigned comment added by AnalysisAlgebra (talk • contribs) 00:21, 29 October 2012 (UTC)

Yes, it's $\mathbb {Z} /5$. It's probably easier to see this if we reverse the order of quotienting (which is valid by the appropriate isomorphism theorem): your ring is $\mathbb {Z} [x]/(2+x)(x^{2}+1)=\mathbb {Z} /((-2)^{2}+1)=\mathbb {Z} /(5).$--80.109.106.49 (talk) 08:11, 29 October 2012 (UTC)

That looks pretty neat but how do you multiply ideals like that? And how did you change the $\mathbb {Z} [x]$ to $\mathbb {Z}$?--AnalysisAlgebra (talk) 10:10, 29 October 2012 (UTC)

Sorry, that was me being sloppy. I meant $(\mathbb {Z} [x]/(2+x))/(x^{2}+1)$. Then $\mathbb {Z} [x]/(2+x)=\mathbb {Z}$ because we're adding a variable $x$, but then setting $x=-2$.--149.148.254.70 (talk) 10:42, 29 October 2012 (UTC)

Thanks. What about $\mathbb {Z} [x]/(4x-5,2x^{2}-4)$ and $\mathbb {Z} [x]/(x^{2}+3,5)$? Using the same technique I get $(\mathbb {Z} [x]/(4x-5))/(2x^{2}-4)=\mathbb {Z} [5/4]/(2(5/4)^{2}-4)=\mathbb {Z} [5/4]/(-7/8)=\mathbb {Q} /(-7/2)$ and $\mathbb {Z} [i{\sqrt {3}}]/(5)$ respectively. Do those make sense?--AnalysisAlgebra (talk) 12:25, 29 October 2012 (UTC)

What map is referred to?

Is there a standard map $R\rightarrow R[x]/(ax-1)$ whose kernel is the set of all elements $b$ of $R$ such that $a^{n}b=0$ for some $n>0$?--AnalysisAlgebra (talk) 10:08, 29 October 2012 (UTC)

How many solutions

How many solutions are there to this equation.

${\frac {1}{a}}+{\frac {2}{b}}-{\frac {3}{c}}=1$

Where a , b and c are all positive integers? 220.239.37.244 (talk) 10:11, 29 October 2012 (UTC)

Infinitely many - consider the cases a = 1 or b = 2. Gandalf61 (talk) 10:23, 29 October 2012 (UTC)

Agreed. The general rule of thumb is that, with 3 variables, you'd need 3 independent equations to define a unique solution. StuRat (talk) 19:20, 29 October 2012 (UTC)

The OP specified that the variables ranged over the positive integers. That makes it a Diophantine equation, and in general that significantly complicates the issue of figuring out how many solutions there are (in fact, the question of whether an arbitrary Diophantine equation has a solution is undecidable). --Trovatore (talk) 01:22, 30 October 2012 (UTC)

Indeed. It is the restriction to positive integer solutions that makes such problems interesting and sometimes very difficult. The OP's equation has two infinite families of solutions (1, 2n, 3n) and (n, 2, 3n), as well as two other "sporadic" solutions outside of these families, (2, 1, 2) and (2, 3, 18). But, for comparison, the superficially similar equation

${\frac {1}{a}}+{\frac {1}{b}}+{\frac {1}{c}}=1$

has only a finite number of solutions in positive integers, which are (2, 3, 6), (2, 4, 4), (3, 3, 3) and their permutations. Gandalf61 (talk) 11:48, 30 October 2012 (UTC)

Yes, but if all terms are positive, then there are a finite numbers of solutions.Naraht (talk) 23:20, 30 October 2012 (UTC)

Universal cover inducing a short exact sequence

Consider the universal cover, p : E→ X, of a space X (assume all spaces are 'nice'). Does this induce some sort of short exact sequence whose kernel is the deck transformation group (i.e. the fundamental group of X), and which involves (subgroups of) the homeomorphism groups of E and X? I'm inferring from a paper I'm reading that there is such a thing. It refers to it as the 'associated short exact sequence' of the covering, so either I'm being dense, or it's a construction I'm not familiar with. Any clues? Thanks, Icthyos (talk) 12:49, 29 October 2012 (UTC)

I think I was just being dense. Any homeomorphism of X lifts to one of E, and this lift is unique up to deck transformations, giving a sequence like the one I mentioned. Right? Icthyos (talk) 16:09, 29 October 2012 (UTC)