Abelian Groups - László Fuchs [Springer]

676 18 Groups in Rings and in Fields
If G is the cotorsion hull of A, then in the exact sequence 0 ! A ˝ A !
G ˝ G ! Œ.G=A ˝ A ˚ ŒG ˝ .G=A/ ! 0 (that is like (18.1)) the direct sum
is torsion-free and divisible. A reference to Theorem 9.2 in Chapter 9 shows that the
map Hom.G ˝ G; G/ ! Hom.A ˝ A; G/ is surjective, completing the proof. ut
F Notes. The pioneering papers: Beaumont [1], Szele [1], Rédei–Szele [1], Beaumont–
Zuckerman [1] (all published around 1950) provide only a superficial analysis of the relation
between the structure of a ring and its additive group, in very special cases. They stimulated
interest in the additive groups of rings, and in the next decade several more substantial papers
were published, including Beaumont–Pierce [1] which initiated a more systematic study of rings
on finite rank torsion-free groups.
Exercises
(1) Describe all (associative) rings on the following groups: Z.p n /; Z.p 1 /; Z.
(2) (a) In a torsion-free ring R with identity 1, .1/ .a/ for each a 2 R:
(b) An unbounded p-ring cannot have identity.
(3) If U; V are fully invariant subgroups (ideals) of R, then their product is
contained in their intersection U \ V.
(4) (Steinfeld) Let R be a ring without divisors of zero. Then either pR D 0 for
some prime p,orR is torsion-free whose typeset is directed upward.
(5) In a torsion-free ring R, the nil radical is a pure ideal.
(6) Let M denote a maximal (left) ideal of the torsion-free ring R. Then either
pR M for some prime p,orR/M is torsion-free divisible.
(7) The ring constructed in Theorem 1.3 has the property that every torsion-free
divisible ring containing a subring isomorphic to the ring on A also contains a
ring isomorphic to the one on D.
(8) Let R be a ring with additive group A such that A 1 D 0. Then the
unique extension ring on the completion QA inherits polynomial identities (in
particular, associativity, commutativity) from R, andisa QZ-algebra. [Hint: QA
is an inverse limit.]
(9) Let A and D be as in Theorem 1.3.
(a) Establish a bijection between the pure left ideals of A and D.
(b) This correspondence preserves primeness.
(c) D can have an identity even if A has none.
(d) If D has an identity, then every left ideal of D is pure.
(e) D has a non-zero nilpotent ideal exactly if A has got one.
(10) A ring on a splitting mixed group need not be a direct sum of a torsion and a
torsion-free ring. [Hint: on hai˚hbi ŠZ ˚ Z.p/ define a 2 D b; ab D ba D
b 2 D 0.]