The question of when certain cyclic flat modules of a ring are injective (and vice versa) is studied. The consequences of the conditions 'flat' and 'injective' on the simple modules of a ring are discussed. Introduction. The object of this paper is to consider the relationships between the injectivity and flatness of cyclic modules of the form R/A where A is a two-sided ideal of the ring R. In general neither the implication R/A is right injective R/A is right flat nor its reverse need be true. However, Ware [11] has recently shown that if R is commutative and A is a maximal ideal, then both implications are true. We generalize this observation to noncommutative rings and indicate some conditions under which the injectivity of R/A implies its flatness. These considerations lead to Baer's notion of pR-completeness and it turns out that, for simple modules over commutative rings, pR-completeness is equivalent to injectivity, thus yielding a new characterzation of commutative von Neumann regular rings. The paper ends with a discussion of pR-completeness and flatness of simple modules over noncommutative rings. Terminology. The terms ring, module and homomorphism will mean ring with unity, unitary right module and right module homomorphism respectively. 'Ideal will mean two-sided ideal. 'If R is a ring and M is a right R-module, M will be said to be I-complete for a right ideal I of R if any homomorphism I -* M can be extended to a homomorphism R -f M. 1. Simple modules. Let A be an ideal of R. 'In this section the connection between the injectivity and flatness of the cyclic module R/A is discussed for the case when A is a maximal right ideal. Received by the editors October 3, 1973 and, in revised form, January 16, 1974. AMS (MOS) subject classifications (1970). Primary 16A30, 16A50, 16A52.
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