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The following question might be a duplicate but I couldn't find anything when I searched just now. (I also would not object hugely if the question gets moved to MSE except that I do not have an account there and so would not be able to respond to comments or accept answers.)

It seems to be folklore that Goursat's lemma from group theory works for other kinds of algebraic structure. An answer to this older MO question gives precise conditions on a category that are sufficient and close-to-necessary for the natural "abstract Goursat lemma" to be valid.

In particular, it seems that this should work for rings. That is, suppose $T$ is a subdirect product of $R\times S$ where $R$ and $S$ are not-necessarily-commutative rings, and $\pi_R:R\times S \to R$, $\pi_S:R\times S \to S$ are the coordinate projections. Then we have ideals $R_0 = \pi_R( T\cap \ker\pi_S) \unlhd R$ and $S_0=\pi_S(T\cap \ker\pi_R)\unlhd S$ and if I am not mistaken the calculation shown in the MO question I linked to shows that $T$ "descends" to the graph of a ring isomorphism $R/R_0 \to S/S_0$.

In some work with my PhD student, we found ourselves needing a special case of this result which we proved by hand. Only later did it occur to me that what we had done should be a special case of a more general known result, and then this led me to look up Goursat's lemma.

My question is this: can someone supply a reference to the literature, such as one of the standard "books people in North American systems read to prefer for qualifier/comprehensive exams", which states Goursat's lemma for rings, and does not deduce it from some general result about e.g. exact Mal'cev categories?

I would prefer that my PhD student does not needlessly reinvent the wheel if there is a clear statement that can be quoted, but I also don't want to force them to suddenly start learning general category-theoretic language when the application we have in mind is much more concrete and specialized.

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    $\begingroup$ I don't have time right now to compare their theorems with your post, but does this paper of Anderson and Camillo help? $\endgroup$
    – Jeremy Rickard
    Commented Sep 13, 2020 at 14:14
  • $\begingroup$ What kind of "more concrete and specialized" mathematics is your student doing? I am just surprised they would be so averse to category theoretic language. $\endgroup$
    – Pedro
    Commented Sep 13, 2020 at 16:48
  • $\begingroup$ @PedroTamaroff I don't really want to get into a discussion about my student's PhD, since it's not really relevant, but (speaking as someone whose own PhD happily employs category-theoretic language and POV, but who works in functional analysis), I am surprised at your surprise. E.g. it really isn't necessary to be thinking of comma categories when doing analysis on RKHS (this isn't the thesis topic, but just for sake of argument) $\endgroup$
    – Yemon Choi
    Commented Sep 13, 2020 at 16:51
  • $\begingroup$ @JeremyRickard Thanks! based on a quick look, it does seem that Theorem 11 part 1(b) would do the job, so if all else fails I might pass that on to my student as a reference that could be used, after encouraging them to work through the proof themselves. $\endgroup$
    – Yemon Choi
    Commented Sep 13, 2020 at 16:58
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    $\begingroup$ @YemonChoi I see. I won't be surprised at your surprise of my surprise, since that may lead us into an odd path. My point is that the requisite language orbits around kernels, subobjects and similar things, which your student may not be terribly unfamiliar with. Best of luck! $\endgroup$
    – Pedro
    Commented Sep 13, 2020 at 16:59

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