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We discuss some open questions regarding the unit sum numbers of free modules of arbitrary infinite rank over commutative rings and, in particular, over principal ideal domains. The unit sum numbers of rational groups are then investigated: the importance of the rational prime 2 being an automorphism of the rational group is discussed and other results are achieved considering the number and distribution of rational primes which are, or are not, automorphisms of the group. We next prove the existence of rational groups with unit sum numbers greater than 2 but of finite value and we estimate an upper bound for the unit sum number of one such group. Knowledge of unit sum numbers for completely decomposable groups is extended. In the final chapter we discuss a new property for abelian groups and modules which we call the involution property. The discussion closely mirrors that of the 2-sum property. Clear results are found for free modules over any principal ideal domain, for rational groups and completely decomposable groups. Complete modules and vector spaces are also discussed.
Meehan, Christopher. Unit sum numbers of abelian groups and modules. Dublin : Dublin Institute of Technology, 2001