Vol. 76, No. 1, 1978

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Decomposing modules into projectives and injectives

Patrick F. Smith

Vol. 76 (1978), No. 1, 247–266

A ring R is called a right PCI-ring if and only if for any cyclic right R-module C either CR or C is injective. Faith has shown that right PCI-rings are either semiprime Artinian or simple right semihereditary right Ore domains. Thus if R1 and R2 are right PCI-rings then R = R1 R2 is not a right PCI-ring unless R1 and R2 are both semiprime Artinian but R has the property that every cyclic right R-module is the direct sum of a projective right R-module and an injective right R-module, and rings with this property on cyclic right R-modules will be called right CDPI-rings. On the other hand, if S is a semiprime Artinian ring then the ring of 2 × 2 upper triangular matrices with entries in S is also a right CDPI-ring. The structure of right Noetherian right CDPI-rings is discussed. These rings are finite direct sums of right Artinian rings and simple rings. A classification of right Artinian right CDPI-rings is given. However the structure of simple right Noetherian right CDPI-rings is more difficult to determine precisely and the problem of finding it reduces to a conjecture of Faith.

Mathematical Subject Classification
Primary: 16A50, 16A50
Secondary: 16A62
Received: 4 May 1977
Published: 1 May 1978
Patrick F. Smith
University of Glasgow
United Kingdom