On the cardinality of smallest spanning sets of rings

Let R=(R,+, · ) be a ring. Then a subset Z of R is called spanning if the R-module generated by Z is equal to the ring R. A spanning set Z is called smallest if there is no spanning set of smaller cardinality than Z. It will be shown that the cardinality of a smallest spanning set of a ring is not always decidable. In particular, a ring will be constructed such that the cardinality of a smallest spanning set of this ring depends on the underlying set theoretic model.

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