A ring R is called right pureinjective if it is injective with respect to pure exact sequences. According to a well known result of L. Melkersson, every commutative Artinian ring is pureinjective, but the converse is not true, even if R is a commutative Noetherian local ring. In this paper, a series of conditions under which right pureinjective rings are either right Artinian rings or quasiFrobenius rings are given. Also, some of our results extend previously known results for quasiFrobenius rings.
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