Definition of direct summand

Question

If M is an R-module ,and if N is a direct sum of M over an indexed set I, then is it correct to write: N= T direct sum M, where T is the direct sum of M over I-{i} , where i in I.

Answer 1

I think you are asking if $$\bigoplus_{j \in I} M = M \oplus \bigoplus_{j \in I \setminus \{i\}} M.$$ The answer is yes. But do you have some doubt about this that you want to clarify?