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 .

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.