Bourbaki

Corollary 1

Tag 05KB, permanent. Cite it as [Bourbaki, Tag 05KB].

Algebra, chapter II, § 1. Modules

Under the conditions of Proposition 7, (i),

Ker(f)=⨁i∈IKer(fi),Im(f)=⨁i∈IIm(fi) \mathrm{Ker}(f) = \bigoplus_{i \in I} \mathrm{Ker}(f_i), \qquad \mathrm{Im}(f) = \bigoplus_{i \in I} \mathrm{Im}(f_i)
and there are canonical isomorphisms
Coim(f)→⨁i∈ICoim(fi),Coker(f)→⨁i∈ICoker(fi) \mathrm{Coim}(f) \to \bigoplus_{i \in I} \mathrm{Coim}(f_i), \qquad \mathrm{Coker}(f) \to \bigoplus_{i \in I} \mathrm{Coker}(f_i)
defined as in no. 5, Corollary to Proposition 5. In particular, for ff to be injective (resp. surjective, bijective, zero), it is necessary and sufficient that each of the fif_i be injective (resp surjective, bijective, zero).

If, for all i∈Ii \in I, we consider a submodule FiF_i of EiE_i, the module ⨁i∈IFi\bigoplus_{i \in I} F_i is a submodule of ⨁i∈IEi\bigoplus_{i \in I} E_i and, by virtue of Corollary 1 to Proposition 7, there is a canonical isomorphism

⨁i∈I(Ei/Fi)→(⨁i∈IEi)/(⨁i∈IFi). \bigoplus_{i \in I} (E_i/F_i) \to \left( \bigoplus_{i \in I} E_i \right) / \left( \bigoplus_{i \in I} F_i \right).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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Transcribed from the printed edition, 1998, Springer. Not translated.