Abelian Groups - László Fuchs [Springer]

700 18 Groups in Rings and in Fields
We shall require a lemma, generalizing a result by Cohn [2].
Lemma 7.7. Let U be the unit group of an integral domain, and assume A is a
multiplicative group containing U such that A=U is torsion-free. Then A, too, is the
unit group of some domain.
Proof. Let R be a domain with U.R/ D U.WeviewA as an extension of U by A=U,
and select a representative a x 2 A in each coset x modulo U. We obtain a factor set
u x;y 2 U as defined by
a x a y D u x;y a xy for all x; y 2 A=U: (18.5)
Define S as an algebra over R with the set fa x g x2A=U as basis such that the basis
elements multiply according to the rule (18.5). Since the u x;y are coming from A,
they satisfy the obligatory associativity conditions to guarantee that S will become
an R-algebra. It is manifest that the multiples ua x of the basis elements a x (where
u 2 U) form a subgroup of U.S/ canonically isomorphic to A.
It still remains to show that S is a domain, and U.S/ contains no elements other
than those of the form ua x . Every torsion-free abelian group admits a linear order
compatible with the group operation. Choose an arbitrary, but fixed linear order
on A=U, and write a non-zero element 2 S in the form D P m
iD1 r ia xi with
0 ¤ r i 2 R, anda xi in the above set of representatives such that x 1 < < x m .
If D P n
jD1 s ja yj with 0 ¤ s j 2 R and y 1 < < y n is another element of
S, then in the product , a x1 y 1
will be the smallest basis element with coefficient
r 1 s 1 u x1 ;y 1
¤ 0. Thus S has no zero-divisors. Since the largest basis element a xm ;y n
in the product will also have a non-vanishing coefficient, it follows that D 1
only if m D 1 D n, and in addition, r 1 ; s 1 are units in R.
ut
Group Rings For group rings, the main difficulty lies in the emergence of the
so-called non-trivial units: units in RŒA that are not products of units in R with
elements of A. There is an extensive literature on the unit groups of groups rings of
a group over a commutative ring. In this volume, we do not discuss group rings.
Example 7.8. Let A Dhai be cyclic of order 5, and let Q .3/ denote the ring of rationals whose
denominators are powers of 3. Then Q .3/ ŒA has non-trivial units. To prove this, consider D
1 C a C a 2 C a 3 C a 4 2 Q .3/ ŒA: Then a D , whence 2 D 5 follows. An easy calculation
shows that 1 2 is a unit with inverse 1
2
9 .
F Notes. Rings with cyclic groups of units were described by Gilmer [1] for finite rings, and
by Pearson–Schneider [1] for infinite rings.
It was G. Higman [Proc. London Math. Soc. 46, 231–248 (1938/39)] who started a systematic
study of unit groups in group rings; he investigated the units of group rings over finite algebraic
extensions of Z. In general, the study of the unit groups in (commutative) group rings is an
interesting area of research. Several significant results are available. The main contributors include
S. Berman, P. Danchev, G. Karpilovsky, W. May, T. Mollov, N.A. Nachev, W. Ullery. From the
point of view of abelian group theory, the results by Danchev are especially interesting: how the
structure of the unit group of the group ring over a p-group depends on the structure of the p-group
(e.g., if the group is totally projective).