In the category of finitely generated modules over a commutative Noetherian ring, the splitting of a short exact sequence can be checked locally at the maximal ideals of the ring. One reference for this is contained in the answer by Jeremy Rickard here [Local property of split exact sequence](https://mathoverflow.net/questions/271588/local-property-of-split-exact-sequence) . 


My question is: Is there a reference of book or paper or some monograph where this fact (possibly along with a proof) appears ? 


I would like to use this in a presentation and this result seems too natural to not have appeared anywhere before in literature. 


Thanks. 



Splitting of short exact sequence in the category of finitely generated modules over a commutative Noetherian ring
Reiner's "Maximal Orders", Theorem 3.20, for instance.


