projective equivalence

Let $R$ be a ring with 1. Two $R$ -modules $A$ and $B$ are said to be projectively equivalent $A\\sim B$ if there existtwo 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:
$\\xymatrix{0\\ar[r]&B_{1}\\ar[r]&P\\ar[r]&A_{1}\\ar[r]&0}$
$\\xymatrix{0\\ar[r]&B_{2}\\ar[r]&Q\\ar[r]&A_{2}\\ar[r]&0}$
with $A_{1}\\sim A_{2}$ , then $B_{1}\\sim B_{2}$ .
4.
Schanuel’s Lemma can be generalized. Given two projective resolutions:
$\\xymatrix{\\ldots\\ar[r]^{p_{3}}&P_{2}\\ar[r]^{p_{2}}&P_{1}\\ar[r]^{p_{1}}&P_{0}%\\ar[r]^{p_{0}}&A_{1}\\ar[r]&0}$
$\\xymatrix{\\ldots\\ar[r]^{q_{3}}&Q_{2}\\ar[r]^{q_{2}}&Q_{1}\\ar[r]^{q_{1}}&Q_{0}%\\ar[r]^{q_{0}}&A_{2}\\ar[r]&0}$
with $A_{1}\\sim A_{2}$ , then $\\operatorname{Ker}(p_{n})\\sim\\operatorname{Ker}(q_{n})$ for all $n\\geq 0$
5.
The concept of projective equivalence between two modules can be generalized to any abelian categories having enough projectives.