Let be a locally finitely presented additive category, and let be a finitely presented pure-injective
object of . We prove that has an indecomposable decomposition if and only if every pure epimorphic
image of is pure-injective if and only if the endomorphism ring of is semiperfect. This extends a
module-theoretic result which generalises the classical Osofsky Theorem.

Key Words: Locally finitely presented category, Krull-Schmidt category, indecomposable decomposition, (completely) pure-injective object, semiperfect ring, semisimple ring, Osofsky theorem.

2010 Mathematics Subject Classification: Primary 18E05,

Secondary 18C35, 16D90.