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Jonsson and HS modules over commutative rings. (English) Zbl 1286.13007

Summary: Let \(R\) be a commutative ring with identity and let \(M\) be an infinite unitary \(R\)-module. (Unless indicated otherwise, all rings are commutative with identity \(1\neq 0\) and all modules are unitary.) Then \(M\) is called a Jónsson module provided every proper submodule of \(M\) has smaller cardinality than \(M\). Dually, \(M\) is said to be homomorphically smaller (HS for short) if \(|M/N|<|M|\) for every nonzero submodule \(N\) of \(M\). In this survey paper, we bring the reader up to speed on current research on these structures by presenting the principal results on Jónsson and HS modules. We conclude the paper with several open problems.

MSC:

13C05 Structure, classification theorems for modules and ideals in commutative rings
13A15 Ideals and multiplicative ideal theory in commutative rings
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