[nubslayer]

>and means you have to define things like the tensor product of abelian groups properly before you can even define a ring

Hmm, someone managed to do it at Wikipedia:

"In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra. It consists of a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication. Through this generalization, theorems from arithmetic are extended to non-numerical objects such as polynomials, series, matrices and functions."[0]

[0] https://en.wikipedia.org/wiki/Ring_%28mathematics%29

[TheDong]

You missed the point of the parent comment.

They were saying that the definition you found, on wikipedia, is better than the definition of "a monoid object in the category of abelian groups" because the definition on wikipedia doesn't require knowing about tensor products etc.

You're proving their point. Wikipedia doesn't define it with that language for exactly the reasons the comment pointed out.

[throwawaymath]

Yes, exactly. That's not category theoretic language, which is why that definition is so approachable. There are far fewer prerequisites to understanding what's going on in that definition.

That's precisely what the parent commenter is getting at.

[nubslayer]

Seems like all language would be "category theoretic".

[throwawaymath]

No, not at all. Here are two equivalent definitions of a ring:

1. A monoid in the Ab-category.

2. A set R which is equipped with addition and multiplication.

The first definition is category theoretic, and requires you to know what abelian groups are and what it means for something to be 1) a category, 2) a monoid, and 3) a monoid in the category of abelian groups.

The second definition is straightforward if you can follow a few axioms and know naive set theory. It is helpful but unnecessary to understand that a ring is an abelian group which also supports multiplication in order to get the second definition. But even if you know this about rings, you'll still need to understand all the heavy lifting behind what the category of abelian groups actually means.

I'm not saying it's not useful. But I am saying one is clearly more accessible than the other, with fewer prerequisites.

[nubslayer]

All those words are in various categories.