Properties of direct summands of modules.

*(English)*Zbl 0659.16016L. Fuchs posed [in his “Infinite abelian groups” I (1970; Zbl 0209.055)] the problem of characterizing the abelian groups in which the intersection of each pair of direct summands is a direct summand. This property (called the summand intersection property or SIP) has been studied, for arbitrary modules, by G. Wilson [Commun. Algebra 14, 21-38 (1986; Zbl 0592.13008)], who also characterized modules with SIP over principal ideal domains. In this paper, modules M are studied with the property that the sum of any pair of direct summands of M is a direct summand of M (summand property or SSP). Characterizations of rings are obtained in terms of the SSP of their modules, and, for a module M, the SSP (or the SIP) of M is studied through properties of \(End_ R(M)\). Furthermore the author studies the conditions when a given module, which can be written as a direct sum of indecomposable modules, has the SSP (or the SIP or both). Some particular cases are investigated.

Reviewer: F.Loonstra

##### MSC:

16D70 | Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) |

16S50 | Endomorphism rings; matrix rings |

16W50 | Graded rings and modules (associative rings and algebras) |

##### Keywords:

direct summands; summand intersection property; SIP; summand property; direct sum of indecomposable modules; SSP
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DOI

##### References:

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[3] | Fuchs L., Infinite abelian groups 1 (1970) |

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[11] | DOI: 10.1080/00927878608823297 · Zbl 0592.13008 |

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