Morita-equivalent modules

Definition

Definition with symbols

Two modules $M_{1}$ and $M_{2}$ over a commutative unital ring are said to be Morita-equivalent if there are projective modules $P_{1}$ and $P_{2}$ such that:

$M_{1}\oplus P_{1}\cong M_{2}\oplus P_{2}$

Modules, modulo the relation of Morita equivalence, give rise to the Morita category.