domain if and only if the torsion submodule is a direct summand whenever it is bounded (M is bounded means rM = 0 for some r in R), . Similarly, a commutative domain is a Prüfer domain if and only if the torsion submodule is a direct summand whenever it is finitely generated . Generalizations More generally, a Prüfer ring is a commutative ring in which every non-zero finitely generated ideal consisting only of non-zero-divisors is invertible (that is, projective). A commutative ring is said to be arithmetical if for every maximal ideal m in R, the localization Rm of R at m is a chain ring. With this definition, an arithmetical domain is a Prüfer domain. Non-commutative right or left semihereditary domains could also be considered as generalizations of Prüfer domains. See also Divided domain References Commutative algebra