Abelian Groups - László Fuchs [Springer]

5 Additive Groups of Regular Rings 691
showing that every p m a is divisible by p 2 m in the principal left ideal Ra. We
derive that p 2 m L D p m L,andp m L is p-divisible. Hence if a 2 L p is of order
p k ,thenh p .p m a/ !, thus for some y 2 R, p m a D p k .ya/ D y.p k a/ D 0,
whence p m L p D 0 is obtained. Hence L p is bounded, so L C D L C p ˚ C for some
subgroup C of L. Notice that p m L D p m C is p-divisible, and division by p in
C is unique. Consequently, C D p m C, and we obtain that L D L p ˚ p m L is a
ring-theoretical direct sum. This completes the proof of (i).
(ii) All that we have to do is to observe that L/T is an epic image of the p-divisible
rings p m L,and-regularity is inherited by surjective images.
ut
Note that the integer m in the preceding theorem depends only on prime p,andis
the same for all left ideals of the -regular ring.
Embedding in Regular Ring with Identity We wish to solve the problem of
embedding of regular rings in rings with identity preserving regularity. We wish to
show that there is always such an embedding. Oddly enough, this is another purely
ring-theoretical question that is answered by taking full advantage of our knowledge
of the additive groups.
To start with, we construct a commutative regular ring M with identity as follows.
For every prime p, take the prime field F p D Z=pZ of characteristic p; let p
denote its identity element. Evidently, F D Q p F p is a commutative regular ring
with identity D .:::; p ;:::/. The quotient F= ˚p F p is a torsion-free divisible
regular ring in which the coset of generates a pure subgroup M= ˚p F p isomorphic
to Q as a ring as well. This M is a regular ring: it contains the regular ring ˚pF p as
an ideal modulo which it is regular.
It might be of interest to point out that the ring F is the completion of M, and its
ring structure is completely determined by M.
Proposition 5.4 (Fuchs–Halperin [1]). Every regular ring is a unital algebra over
the commutative regular ring M.
Proof. Let R be a regular ring. To define the action of x 2 M on a 2 R, we write
x D .:::;x p ;:::/ with x p 2 F p , and notice that, by construction, there is a rational
number mn 1 .m; n 2 N/ such that nx p m mod p for almost all primes p. Select
a finite set fp 1 ;:::;p k g of primes which includes all prime divisors of m; n as well
as the primes for which the last congruence fails to hold. With such a set of primes,
we make a ring-decomposition
M D F p1 ˚˚F pk ˚ M 0 ; (18.3)
where M 0 is an ideal of M such that multiplication by each p i .i D 1;:::;k/ is an
automorphism of M 0 . Accordingly, we have
x D x p1 CCx pk C x 0 .x pi 2 F pi ; x 0 2 M 0 /: