Radicals commuting with cartesian products

Abstract

For any radical R of abelian groups which does not commute with arbitrary cartesian products we define the norm kRk to be the least cardinal for which there exists a family, of this size, of groups G a such that R G a j RG a . This norm kRk is always regular. Assuming GCH, we construct reduced products G to show that every regular cardinal k which is not greater that any weakly compact cardinal is the norm of a suitable group radical R G .

How rigid are reduced products?

“…This sequence of sets is then used to construct test subgroups of Z that correspond to our smaller test groups T J in Proposition 4.5. Wald's method was also used in [6].…”

“…Hence, it is very natural to study the case when GCH does not hold or if we are above the first weakly compact cardinal. An inspection of the proof in [6] shows that GCH was needed to overcome a cardinal restriction in a nice result due to Wald [27]; see also Corollary 6.2. He proved the following two theorems [27,Theorems A,B].…”

“…For any radical R of abelian groups which does not commute with arbitrary cartesian products we define the norm R to be the least cardinal for which there exists a family, of this size, of groups G such that R G = RG , see [14]. This norm R , if it exist, is always regular, see [6] and clearly, t = ℵ 0 . Eda [10] showed that satisfies ℵ 1 2 ℵ 0 .…”

“…This follows from a quite general theorem (see [9]) to the effect that every nonzero slender group G satisfies R G = ℵ fm , where ℵ fm denotes the first measurable cardinal, or R G = ∞ if the universe admits no measurable cardinal. Assuming GCH, Corner and Göbel [6] constructed reduced products G to show that every regular cardinal which is not greater than any weakly compact cardinal is the norm of a suitable group radical R G . Hence, it is very natural to study the case when GCH does not hold or if we are above the first weakly compact cardinal.…”

The Chase radical and reduced products

“…For a torsion-free abelian group G, ν(G) is then defined as ν(G) = {Ker(ϕ) | ϕ : G −→ X with X ℵ 1 -free}. Radicals in general and, in particular group radicals, got much attention and were studied extensively (see [2] and the references in there). However, the Chase radical plays a distinguished role since it tests the ℵ 1 -freeness of torsion-free abelian groups, in other words, a torsion-free abelian group G is ℵ 1 -free if and only if ν(G) = 0.…”