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I'm puzzled by the definition of ideals generated by a subset of a ring in Aluffi, Algebra: Ch 0.

The previous chapter on groups is (for an algebra book of this level) quite categorical in spirit. In particular, it gives definitions and proofs in terms of universal properties whenever possible. For example, given a subset $A$ of a group $G$, the subgroup generated by $A$ is defined as the image of the unique group homomorphism $F(A) \to G$ where $F(A)$ is the free group on $A$.

This perspective returns when the submodule generated by a subset of a module is defined using the universal property of free modules. Meanwhile, however, the definition of the ideal generated by a subset of a ring is more traditional. It is essentially defined to be the smallest ideal of the ring containing the subset.

Since the book makes so much effort to give element-free definitions in terms of universal properties, I'm wondering whether such an approach is possible for defining ideals generated by subsets, and if so, why Aluffi is avoiding it. I suppose one cannot hope to straightforwardly mimic the group or module case as the images of ring homomorphisms are not in general ideals.

David M
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  • Well, if $R$ is a unital commutative ring, then the notions of "submodule"of $R$" and "ideal of $R$" coincide, so you may apply the point of view of submodule generated by a subset using free modules (even if I don't really see the point of defining subgroups/submodules/subgroups generated by a subset by a universal property. – GreginGre Jun 29 '24 at 10:34
  • The book does assume that all rings are unital. But if $R$ is not commutative? It's possible though that he just wants to get on to the commutative case as quickly as possible. – David M Jun 29 '24 at 11:00
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    @DavidM: You might want to look at a similar question that I asked a few months ago. The answer by Martin Brandenburg (a very categorically minded person) suggests that it is actually probably not right from a conceptual point of view to think of subwhatevers as being defined in terms of free objects. (This is not to say that the "free object" perspective is never useful, but perhaps it should not be the primary way of thinking about things.) – Joe Jun 29 '24 at 23:35

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It's a special case of the definition of the submodule generated by a set: for left ideals you take left submodules, for right ideals you take right submodules, and for two-sided ideals you take two-sided submodules.

I am not Aluffi but I will speculate that the reason he treats this case differently is that there is a universal computation happening but it isn't happening in the category of rings. The universal thing in the category of rings would define the subring generated by a set, rather than the ideal. To define the ideal generated by a set categorically you have to leave the category of rings (and enter the category of modules, as above), which Aluffi might have considered unsatisfying or not worth getting into.

In the case of two-sided ideals an alternative approach is the following. If $S$ is a subset of a ring $R$ then we can consider the universal ring map $f : R \to R'$ such that $f(S) = 0$. This map always exists, and then we can define the ideal generated by $S$ to be its kernel. (However, note that this kernel also does not make sense in the category of (unital) rings! This can be fixed by taking the kernel pair, which defines a congruence. This contains the same information as the kernel of $f$ but stays entirely in the category of rings.)

Qiaochu Yuan
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