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