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Problem statement: If R is a ring such that a^2 = a, for all a ∈ R, then show that R is a commutative ring.

The answer: enter image description here

I have NO idea why one should go about doing whatever the proof does. How should I hone the skills needed to tackle such problems?

More importantly, will honing my skills in such algebraic manipulations help me further down the line in advanced college algebra, even research?

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    $\begingroup$ This example is... an (iconic) example, of a phenomenon that we'd probably never imagine... but "turn out to be" genuine. I think there's no good "general" way to anticipate all such things, since, after all, mathematics is not just an extrapolation of middle-school algebra + calculus. :) $\endgroup$ Apr 2 at 21:12
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    $\begingroup$ Probably, if you knew the result about groups, first, it would have been easier to find this argument. In a group, $G$ with $x^2=1$ for all $x\in G,$ then $G$ is commutative. The proof there is $1=(ab)^2=a^2b^2,$ and then, multiplying on the left by $a^{-1}$ and the right by $b^{-1},$ we get $ba=ab.$ This isn't exactly the same, of course, but it seems related., somehow. $\endgroup$ Apr 2 at 22:16
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    $\begingroup$ I’m voting to close this question because it is a highly duplicated question wrapped in a personal advice question. $\endgroup$
    – rschwieb
    Apr 3 at 2:47
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    $\begingroup$ Related: Trouble with 'elementary questions' as a beginner math student (Soft Question). In particular, see my answer, which could be described as saying it's more about the journey than the destination, a viewpoint I think also applies to the computations you're asking about. $\endgroup$ Apr 3 at 4:54
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    $\begingroup$ Does this answer your question? Trouble with 'elementary questions' as a beginner math student (Soft Question) $\endgroup$
    – peterwhy
    Apr 3 at 14:29

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I think it's arguable how much of algebra "boils down" to algebraic manipulation, sometimes plenty, often very little. In general though, the algebraic manipulation often has a thought process behind it, which is often widely applicable and worth learning.

There are many examples, especially in commutative algebra, where the calculation/manipulation can often be recast geometrically, which certainly helps one see where the arguments come from.

Not quite geometrically, but in this instance, in this example we have a very strong axiom (idempotence of every element), for a ring, so we should check how it interacts with other aspects of the ring axioms. For instance, the first is additivity and distributivity, and the second can be thought as, what does this imply for the characteristic of $R$?

One way to also approach this problem (a priori) is to pretend you have a subalgebra of matrices with this property. This then gives that every element is a projection, even when you add projections together, or multiply them, this is pretty weird! For instance, what do the eigenvectors say, etc.

It is important for students to learn how to write and come up with these proofs though, as in research one often has to "play" with the assumptions and see if anything falls out, to develop intuition for the problem.

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  • $\begingroup$ I don't understand the subalgebra of matrices stratagem. Please explain. Thank you! $\endgroup$
    – S_M
    Apr 2 at 21:09
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    $\begingroup$ The strategy being that many interesting rings can show up as subalgebras of matrices, so given as arbitrary ring, if it were matrices, what would this imply? In this instance, if you have intuitions about projections in linear algebra, this might guide your intuition for the problem. This is just one strategy though, some people might like finite rings, or commutative rings, or other classes of rings for test cases. $\endgroup$
    – Chris H
    Apr 2 at 21:18

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