On the cardinality of smallest spanning sets of rings

Abstract

Let R = (R, +, ·) be a ring. Then Z ⊆ R is called spanning if the R-module generated by Z is equal to the ring R. A spanning set Z ⊆ R 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 R is not always decidable. In particular, a ring R = (R, +, ·) will be constructed such that the cardinality of a smallest spanning set Z ⊆ R depends on the underlying set theoretic model.
Mathematics Subject Classification (2000): 13A18, 03E35
Key words: Spanning sets of rings, dominating number, bounding number

Quaestiones Mathematicae 26(2003), 321325