Morita Equivalence

A relation between rings (or algebras) R and S asserting that their categories of (right) modules Mod-R and Mod-S are equivalent as categories; Morita equivalence implies that R and S present the same module-theoretic behavior and many homological invariants coincide despite the rings not being isomorphic.

An equivalence relation on rings or algebraic objects asserting that two rings are Morita equivalent when their categories of (right or left) modules are equivalent as categories, so they share the same representation-theoretic behavior and module-theoretic invariants.

An equivalence relation between rings (or algebras) asserting that their categories of modules (typically left modules) are equivalent as abelian categories via exact functors that preserve direct sums; two rings are Morita equivalent when they present the same module-theoretic representation theory.