Rings whose indecomposable modules are pure-projective or pure-injective

Let P be the class of rings for which every indecomposable right module is pure-projective or pure-injective. When R is a Noetherian local commutative ring of maximal ideal P, it is proven that R is an element of P if and only if R is either an artinian valuation ring or a discrete valuation domain of rank one with rank (R) <= 2 where R is the completion of R in its P-adic topology. Let R be a commutative ring. Then R is an element of P if and only if R is a clean arithmetical ring with R-P is an element of P for each maximal ideal P of R. Moreover, R is a semi-perfect ring when it is Noetherian. Some examples of commutative rings of the class P are given.