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Modules with the summand intersection property. (English) Zbl 0592.13008
If F is a free module over a principal ideal domain (PID), then the intersection of any two summands of F is a summand. In this paper the corresponding question for modules over a ring arises: an R-module M has the SIP (resp. SSIP) if the intersection of two summands is again a summand (resp. if the intersection of any number of summands is a summand).
We give some results of section I: (1) M has SIP iff for every pair of summands S and T with projection $$\pi : M\to S,$$ the kernel of the restriction $$\pi$$ $$| T$$ is a summand. If M has SIP and $$S\oplus T$$ is a summand of M, then the kernel of any morphism from S to T is a summand. - (2) Let $$M=\oplus M_ i$$ be a direct sum of fully invariant submodules $$M_ i$$; then M has SIP (resp. SSIP) iff each $$M_ i$$ has SIP (resp. SSIP). - (3) All projective $$R$$-modules have SIP iff R is hereditary. - (4) The following are equivalent for a ring $$R$$: (i) $$R$$ is semi-simple; (ii) all $$R$$-modules have SSIP; (iii) all $$R$$-modules have SIP; (iv) all injective modules have SIP.
Section II gives a fairly complete description of modules with SIP over a noetherian domain having a nonzero injective submodule. - Section III classifies torsion modules with SIP over Dedekind domains.
Reviewer: F.Loonstra

##### MSC:
 13C05 Structure, classification theorems for modules and ideals in commutative rings 13C13 Other special types of modules and ideals in commutative rings
##### Keywords:
summand intersection property; SIP; SSIP
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##### References:
 [1] DOI: 10.1080/00927878308822908 · Zbl 0515.20034 [2] DOI: 10.1112/plms/s3-15.1.680 · Zbl 0131.02501 [3] Fuchs L., Infinite Abelian Groups 2 (1970) · Zbl 0209.05503 [4] Kaplansky I., Infinite Abelian Groups (1969) · Zbl 0194.04402 [5] Kaplansky I., Commutative Rings (1974)
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