Let $R$ be a ring with 1. Two $R$ -modules $A$ and $B$ are said to be projectively equivalent $A\sim B$ if there exist two projective $R$ -modules $P$ and $Q$ such that $$A\oplus P\cong B\oplus Q.$$

Remarks.

1. Projective equivalence is an equivalence relation.
2. Any projective module is projectively equivalent to the zero module.
3. (Schanuel's Lemma). Given two short exact sequences:
   
   
   with $A_1\sim A_2$ , then $B_1\sim B_2$ .
4. Schanuel's Lemma can be generalized. Given two projective resolutions:
   
   
   with $A_1\sim A_2$ , then $\operatorname{Ker}(p_n)\sim\operatorname{Ker}(q_n)$ for all $n\geq0$
5. The concept of projective equivalence between two modules can be generalized to any abelian categories having enough projectives.