A separative ring is one whose finitely generated projective modules satisfy the property A ⊕ A ≅ A ⊕ B ≅ B ⊕ B ⇒ A ≅ B. This condition is shown to provide a key to a number of outstanding cancellation problems for finitely generated projective modules over exchange rings. It is shown that the class of separative exchange rings is very broad, and, notably, closed under extensions of ideals by factor rings. That is, if an exchange ring R has an ideal I with I and R/I both separative, then R is separative.
Ara, P., Goodearl, K. R., O'Meara, K. C., & Pardo, E. (1998). Separative cancellation for projective modules over exchange rings. Israel Journal of Mathematics, 105, 105-137. https://doi.org/10.1007/BF02780325