Abelian Groups - László Fuchs [Springer]

698 18 Groups in Rings and in Fields
left adjoint functor G which is defined by letting the group ring over the integers
correspond to a group, i.e. GW A 7! ZŒA. Indeed, for each group A and each ring R,
there is a natural bijection between the set of homomorphisms A ! U.R/ and the
set of identity preserving ring maps ZŒA ! R.
Example 7.1. (a) The units in Z=p k Z, for a prime power p k , are the cosets coprime to p. Number
theory tells us that this is a cyclic group of order '.p k / D p k p k 1 generated by any primitive
root mod p k (' is the Euler totient function).
(b) The units of the ring Z=nZ are represented by the residue classes that are coprime to n. The
unit group is a direct product of cyclic groups, its order is '.n/. It is cyclic if and only if
n D p k ;2p k for odd primes p,orelsen D 2; 4.
Example 7.2. The unit group U.J p / is isomorphic to Z.p 1/ ˚ J p for every odd prime p. This
will follow from Theorem 8.6.
Example 7.3. If R is the ring of integers in a finite algebraic extension of Q, thenbyDirichlet’s
theorem on units, U.R/ is the direct product of a finite cyclic group of even order and a finitely
generated free group.
Before listing some informative facts about unit groups, we repeat: all rings to be
considered are commutative with 1, even if this is not stated explicitly.
(A) The unit group of a cartesian product of rings is the cartesian product of the
unit groups.
(B) The polynomial ring S D RŒx i i2I over a domain R with any number of
indeterminates x i satisfies U.S/ Š U.R/. In fact, only a constant polynomial
may be a unit.
(C) The situation for formal power series rings is completely different. Let S D
RŒŒx be the power series ring over a domain R in a single indeterminate x.
Then we have a direct product U.S/ Š U.R/E where E is the multiplicative
group of all power series with constant term 1. This follows from the fact that
every unit of S is the product of a unit of R andanelementofE.
(D) We now consider the localization R S of R at a semigroup S; suppose S contains
only non-zero divisors, and 1 2 S. ThenU.R S / Š U.R/G where G denotes
the group of quotients of S.
(E) Let M be an R-module. The ‘idealization’ R.M/ is a ring with additive group
R ˚ M with the following rules of operations:
.r; a/C.s; b/ D .rCs; aCb/; .r; a/.s; b/ D .rs; rbCsa/
.r; s 2 R; a; b 2 M/:
Then .1; 0/ is the identity of R.M/,and.r; a/ is a unit if and only if r 2 U.R/.
As a result, we have U.R.M// Š U.R/ M.
An immediate consequence of (E) (with R D Z) is the following theorem.
Theorem 7.4. For every group A, there exists a ring with unit group isomorphic to
the additive group Z.2/ ˚ A.
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(F) Of special interest is the role the Jacobson radical J of R plays in connection
with U.R/. Forallr 2 J, 1 C r 2 U.R/, andifJ is regarded as a group under
the ‘circle’ operation r ı s D r C s C rs, then the correspondence r 7! 1 C r
is an isomorphism of .J; ı/ with a subgroup of U.R/.