Modules with the summand intersection property.

*(English)*Zbl 0592.13008If 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.

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 |

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DOI

##### References:

[1] | DOI: 10.1080/00927878308822908 · Zbl 0515.20034 |

[2] | DOI: 10.1112/plms/s3-15.1.680 · Zbl 0131.02501 |

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