On a question of Faith in commutative endomorphism rings

Abstract

Given a commutative ring R R , let Q ( R ) Q(R) denote its maximal ring of quotients and, for any ideal I I of R R , let End ⁡ ( I ) \operatorname {End} (I) denote the ring of R R -endomorphisms of I I . It is known that if Q ( R ) Q(R) is a self-injective ring then End ⁡ ( I ) \operatorname {End} (I) is commutative for each ideal I I of R R . Carl Faith has asked if the converse holds. It does if R R is either Noetherian or has no nontrivial nilpotent elements but here we produce an example to show that it does not hold in general.