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Ideals of direct products of rings

Department of Algebra and Geometry, Faculty of Science, Palacký University Olomouc, 17. Listopadu 12, 771 46 Olomouc, Czech Republic

E-mail Address: ivan.chajda@upol.cz

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Günther Eigenthaler

TU Wien, Faculty of Mathematics and Geoinformation, Institute of Discrete Mathematics and Geometry, Wiedner Hauptstraße 8-10, 1040 Vienna, Austria

E-mail Address: guenther.eigenthaler@tuwien.ac.at

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Helmut Länger

TU Wien, Faculty of Mathematics and Geoinformation, Institute of Discrete Mathematics and Geometry, Wiedner Hauptstraße 8-10, 1040 Vienna, Austria

Department of Algebra and Geometry, Faculty of Science, Palacký University Olomouc, 17. Listopadu 12, 771 46 Olomouc, Czech Republic

E-mail Address: helmut.laenger@tuwien.ac.at

Corresponding author.

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https://doi.org/10.1142/S1793557118500948Cited by:4 (Source: Crossref)

Abstract

It is known that an ideal of a direct product of commutative unitary rings is directly decomposable into ideals of the corresponding factors. We show that this does not hold in general for commutative rings and we find necessary and sufficient conditions for direct decomposability of ideals. For varieties of commutative rings, we derive a Mal’cev type condition characterizing direct decomposability of ideals and we determine explicitly all varieties satisfying this condition.

Communicated by J. Koppitz

Keywords:

AMSC: 13A15, 16R40, 08B05

References

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