About direct sum/product and subdirect product of rings

Question

I'm studying ring theory and I'm stuck in the topic of direct product of rings. If someone can help me, here are my questions: Is there something called direct sum in the category of unital rings?, if the answer is no, what is the coproduct of (unital) rings? Also, is valid the notion of subdirect product in rings with unity? Finally, is there a book where I can study these topics?

I only have two books of ring theory where the authors study direct sums/products of rings, but they don't assume that rings have $1$, so I'm having trouble understanding the definitions. Thanks in advance.

Yes there is. Direct sum is exactly what you expect it to be. — Adam Hughes, Oct 17 '16 at 16:11
@AdamHughes but the natural inclusion map of a ring into its direct sum isn't a monomorphism. — Xam, Oct 17 '16 at 16:18
Huh? if you have
$$A\oplus\bigoplus_{n=1}^M A_i= R$$

then the map $A\to R$ with $a\mapsto (a, 0,0,0,\ldots , 0)$ seems injective to me. Also "its direct sum" doesn't make sense since individual rings don't have direct sums, that's something about an aggregate of rings. — Adam Hughes, Oct 17 '16 at 16:24
@AdamHughes but $a\rightarrow (a,0,\ldots ,0)$ isn't a homomorphism of unital rings. And I'm talking about a direct sum/product and subdirect product of a family of arbitrary rings. — Xam, Oct 17 '16 at 16:33
Possible duplicate of How to construct the coproduct of two (non-commutative) rings — Mees de Vries, Oct 17 '16 at 16:53
@MeesdeVries I'm not asking about only coproduct of rings, but also about the existence of direct sum and subdirect product/sum (if direct sum exists) of rings. — Xam, Oct 17 '16 at 17:38
@AdamHughes For inifinite coproducts, I think the situation for rings is more complicated than the naive direct sum. — rschwieb, Oct 17 '16 at 19:56
@Charter Actually I have only met the definition of "subdirect product" in the category of rings. Where else have you met it? — rschwieb, Oct 17 '16 at 19:57
@rschwieb https://en.wikipedia.org/wiki/Subdirect_product my problem it's that I don't know universal algebra at all :( — Xam, Oct 17 '16 at 20:39
@rschwieb absolutely correct, I should have noted "finite" coproducts. — Adam Hughes, Oct 17 '16 at 20:55
The notion of subdirect product is valid for unital rings, because unital rings can be defined purely by equations, i.e. they form a variety in the UnivAlg sense. — Berci, Oct 17 '16 at 23:08
@Berci interesting... But I'm having a hard time seeing the connection with the identity. It seems like the same definition works for rngs too — rschwieb, Oct 18 '16 at 01:57

score 1 · Accepted Answer · edited Jun 12 '20 at 10:38

Is there something called direct sum in the category of unital rings?

The earliest use of the phrase "direct sum" that can be located (today) on MathSciNet is in the paper:

Subrings of Direct Sums by Neal H. McCoy, American Journal of Mathematics, Vol. 60, No. 2 (Apr., 1938), pp. 374-382.

The definition given for a direct sum of rings is: "If $K$ is any ring, a direct sum of rings $K$ is understood to be the ring of all functions with values in $K$, defined on some finite or infinite set $M$." Under this definition, $R$ is a direct sum of copies of $K$ if it is a subring of $K^M$ for some $M$.

Within a short time "direct sum" was being used as a synonym for "direct product" in the case of finitely many factors, e.g. in

Dyer-Bennet, John A note on finite regular rings. Bull. Amer. Math. Soc. 47, (1941). 784-787.

Nowadays it seems that ring theorists still use direct sum as a synonym for direct product in the case where there are finitely many factors.

if the answer is no, what is the coproduct of (unital) rings?

For an equationally definable class of algebras, the forgetful functor creates products, which implies that the definition of product in such a category does not change when you enlarge or shrink the category. But the forgetful functor does not create coproducts. The construction of coproducts depends on the context. In the category of commutative unital rings, the coproduct of $R$ and $S$ is $R\otimes_{\mathbb Z} S$. In the category of all unital rings it is something different.

One way to deal with coproducts in equational definable categories of algebras is through presentations. If $A$ is presented by $\langle G\;|\;R\rangle$, $B$ is presented by $\langle H\;|\;S\rangle$ and $G\cap H=\emptyset$, then $A\sqcup B$ has presentation $\langle G\cup H\;|\;R\cup S\rangle$.

Also, is valid the notion of subdirect product in rings with unity?

Yes, a subobject of a product is subdirect if it projects onto any coordinate.

is there a book where I can study these topics?

Bergman, Clifford, Universal algebra. Fundamentals and selected topics. Pure and Applied Mathematics (Boca Raton), 301. CRC Press, Boca Raton, FL, 2012.

Dauns, John, Modules and rings. Cambridge University Press, Cambridge, 1994.

"Direct sum" and "direct product aren't "used as synonyms", they are two distinct notions, and they happen to coincide for a finite number of rings. — Najib Idrissi, Oct 18 '16 at 13:38

About direct sum/product and subdirect product of rings

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