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Stream: theory: mathematics

Topic: Ideals of the coproduct of commutative rings

Ryan Schwiebert (Dec 04 2025 at 03:20):

One can characterize the ideals of a finite product of rings (commutative or not) quite nicely. Infinite products behave less well.

For commutative rings, the coproduct of two rings is just their $\mathbb Z$ tensor product, and I assume that works for small coproducts too. But I don't believe I've ever heard, nor can I find any basic information on the ideals of the tensor product of rings. That could be either because someone thinks it is exceedingly simple or because the answer is that there's no nice answer.

Does anyone have information worth knowing about the ideals of, at least, $R_1\otimes_\mathbb Z R_2$?

Ryan Schwiebert (Dec 04 2025 at 03:28):

One other minor request along these lines: can anyone recommend a good text that covers a lot about CRing? I think the resources I currently have don't go into detail about it.

John Baez (Dec 04 2025 at 10:47):

I think it's hard to say a lot about ideals in $R_1 \otimes_{\mathbb{Z}} R_2$ in terms of the ideals of the commutative rings $R_1$ and $R_2$. Consider the example where $R_1 = R_2 = \mathbb{C}[x]$ is the ring of polynomials in one complex variable. $\mathbb{C}[x]$ is a principal ideal domain so every ideal is generated by some polynomial $P$, and every such polynomial factors uniquely into linear factors, so ideals in $\mathbb{C}[x]$ correspond bijectively to finite multisets of points in the plane. In other words: an ideal is just a bunch of points in $\mathbb{C}$: it records the multiset of zeros of a polynomial, where we count zeros with multiplicity.

But ideals of $\mathbb{C}[x] \otimes_{\mathbb{Z}} \mathbb{C}[x] \cong \mathbb{C}[x,y]$ are a lot more complicated. Not every ideal is principal now, but even looking at just the principal ideals, we get tons, not all coming in any obvious way from the ideals in $\mathbb{C}[x]$.