Vol. 26, No. 2, 1968

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A family of functors defined on generalized primary groups

Ray Mines, III

Vol. 26 (1968), No. 2, 349–360

Let G denote an abelian group; G is called a generalized p-primary group if qG = G for all primes qp. Let α be an ordinal, and let δ : G Eα satisfy the following four conditions: (1) Eα is pα Ext-injective, (2) pαEα = 0, (3) δG is pα-pure in Eα, (4) ker δ = pαG. Define pαG to be that subgroup of Eα such that pα(Eα∕δ(G)) = pαG∕δ(G). If α is a limit ordinal, let Lα(G) = lim
←−β<αG∕pβG. Let

U(G ) = Ext(Z(p∞ ),G ) and  Uα(G ) = U(G )∕pαU (G).

Then we have the following pα-pure containments: G∕pαG δ(G) Uα(G) pα(G) LαUα(G), whenever α is a countable limit of lesser hereditary ordinals. We have pαG = Uα(G) for all groups G if and only if pα Ext is hereditary. From this we obtain a new proof of the fact that pα Ext is hereditary when α is a countable limit of lesser hereditary ordinals. We also obtain an example of a cotorsion group G such that G∕pαG is not equal to Lα(G), thus refuting a conjecture of Harrison. A group G is called generally complete if Lα(G)∕δ(G) is reduced for all limit ordinals α. A generalized p-primary group G is generally complete if and only if it is cotorsion.

Mathematical Subject Classification
Primary: 20.50
Received: 1 August 1966
Published: 1 August 1968
Ray Mines, III