Abelian Groups - László Fuchs [Springer]

692 18 Groups in Rings and in Fields
Now R has a decomposition R D T p1 ˚˚T pk ˚ R 0 ; corresponding to (18.3).
If we write accordingly a D a p1 CCa pk C a 0 with a pi 2 T pi ; a 0 2 R 0 ,thenwe
can define
xa D x p1 a p1 CCx pk a pk C mn 1 a 0 : (18.4)
Take into account that T p is an F p -vector space, so that x p a p makes sense for every
p, and so does mn 1 a 0 by virtue of the choice of the primes (multiplication by n
is an automorphism on R 0 ). We still have to convince ourselves that xa does not
change if we select a larger set of primes or use a different form for mn 1 . But this
follows right away from our selection of primes which guarantees that each x p with
p …fp 1 ;:::;p k g acts on F p as a multiplication by mn 1 . We leave it to the reader to
check the algebra postulates to conclude that R is an M-algebra, indeed. ut
By taking full advantage of the last result, it becomes easy to verify the following
theorem.
Theorem 5.5 (Fuchs–Halperin [1]). Every regular ring can be embedded as an
ideal in a regular ring with identity.
Proof. Given a regular ring R (without identity), we form the set of pairs .x; r/ 2
M R, and define operations as usual to make it into a ring R : addition: .x; r/ C
.y; s/ D .x C y; r C s/, and multiplication:
.x; r/ .y; s/ D .xy; xs C yr C rs/ with x; y 2 M; r; s 2 R:
Then .; 0/ is the identity in R ,andr 7! .0; r/.r 2 R/ embeds R in R as an ideal.
As both the ideal and the factor ring are regular, so is R .
ut
F Notes. We refer to additional literature to the embedding problem by various authors, see,
e.g., Jackett [1] where more information is given about the embedding between the direct sum
and the direct product. Proposition 5.4 and Theorem 5.5 are excellent examples to demonstrate
the relevance of the additive group in rings. No method of proof is known to me that avoids the
additive structure.
Results on -regular rings, similar to those in Proposition 5.4 and Theorem 5.5, were proved by
Fuchs–Rangaswamy [1], provided the additive group satisfies a necessary condition. The methods
were similar. For additional results, consult Feigelstock [Fe], where more additive structures are
examined. Feigelstock [2] deals with the additive groups of self-injective rings.
Exercises
(1) A group is the additive group of a boolean ring (every element is idempotent) if
and only if it is an elementary 2-group.