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Ideals of direct products of rings. (English) Zbl 1396.13003

Summary: 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 find necessary and sufficient conditions for the 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.

MSC:

13A15 Ideals and multiplicative ideal theory in commutative rings
08B05 Equational logic, Mal’tsev conditions
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References:

[1] Adamson, I. T., Rings, Modules and Algebras, (1971), Oliver and Boyd, Edinburgh · Zbl 0226.16003
[2] Chajda, I.; Länger, H., Commutative rings whose ideal lattices are complemented, Asian-European J. Math., 12, 1950039-1-12, (2019) · Zbl 1417.13001
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[4] Lambek, J., Lectures on Rings and Modules, (1976), Chelsea Publications, New York · Zbl 0143.26403
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