What does "composé direct" mean in mathematical French?

2011-11-27T19:47:33Z

In EGA IV, Lemma 6.14.1.1, the first sentence is

Soit $R$ un anneau composé direct d'un nombre fini de corps.

I'm guessing this means

Let $R$ be a ring that is the direct product of a finite number of fields,

but I'm not positive. For one thing, "produit direct" seems to be the more usual term for "direct product." I've tried looking up "composé direct" on Google and on Wikipedia.fr, and I have not been able to find a definitive reference. I'm also confused by the fact that "direct" appears to be an adjective in the French, but I can't see what noun it might be modifying if my translation is correct. Google translate gives

Let R be a ring composed of a direct finite number of bodies.

Obviously, "bodies" should be "fields," but it still does not seem to be helpful. Thus, my question:

Question: What does "composé direct" mean in this context? In particular, why is "direct" being used as an adjective?

Secondary Question: Is there a good reason that "composé direct" is being used here rather than another, more searchable term such as "produit direct"?

Answer by quid for What does "composé direct" mean in mathematical French?

2011-11-27T21:22:53Z

In Bourbaki 'composé direct' is the analog for algebras of 'somme direct' (direct sum) for modules (not direct product of modules). As pointed out by OP this makes sense as Bourbaki's algbras do not need to have a multiplicative unit [see also below]. In particular, while for the precise situation of OP 'direct product' would work as translation (regarding the meaning) this is (only) due to the fact that the number of fields is finite; if it were infinite the 'composé direct' would not be the direct product; but what I would call the direct sum of the fields. [As an aside: it seems to me, though I am in no way an authority on this, that Bourbaki uses rather just 'produit' and not 'produit direct.']

In more detail in Théorie des ensembles (E IV.32, §3) after the existance of a certain 'application universel' is established (under certain conditions) it is said that this applies to groups, modules and algebras. And

... est appelé produit libre des $A_i$ dans le cas des groupes, sommes directe dans le cas des modules, composé direct dans le cas des algèbres.

Meaning: ... is called 'produit libre' of the $A_i$ in the case of groups, 'somme direct' (direct sum) in the case of modules, 'composé direct' in the case of algebras.

In an earlier version of this answer, I possibly caused confusion, by saying this is also used for rings and fields. What I meant is that, as in the given passage or, e.g., in Bourbaki's Algèbre Commutative one can find the 'composé direct' terminology applied when the constituing structures are fields or rings. However, the quoted definition/terminology is just for algebras so that for this construction the ring or the field is merely considered as an algebra and the resulting structure (thus) does not have to be a field or ring.

Moreover, it seems to me that the property of this above mentioned 'application universel' is that of the coproduct in the relevant category. However, since I am not too firm in all category stuff and in particular rather unfamiliar with certain terminology of Bourbaki used there I do not vouch for this. (The reference is here, corrections welcome.) Yet, it is not the product; for various reasons including the soft ones that it is dicussed earlier and this would not make sense since for modules this precise construction is 'somme direct' and this certainly means only finitely many 'non-zero coordinates.'

What does "composé direct" mean in mathematical French? - MathOverflow

What does "composé direct" mean in mathematical French?

Answer by quid for What does "composé direct" mean in mathematical French?