Martindale ring of quotients
This ring of quotients was introduced in [a6] as a tool to study prime rings satisfying a generalized polynomial identity. Specifically, let be a prime ring (with
) and consider all pairs
, where
is a non-zero ideal of
and where
is a left
-module mapping. One says that
and
are equivalent if
and
agree on their common domain
. This is easily seen to yield an equivalence relation, and the set
of all equivalence classes
is a ring extension of
with arithmetic defined by
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Here, indicates the mapping
followed by the mapping
.
One can show (see [a12]) that the left Martindale ring of quotients is characterized as the unique (up to isomorphism) ring extension
of
satisfying:
1) if , then there exists a
with
;
2) if and
, then
; and
3) if and
, then there exists a
with
for all
. As a consequence, if
is simple, then
. In any case,
is certainly a prime ring. The right Martindale ring of quotients
is defined in an analogous manner and enjoys similar properties.
Again, let be a prime ring and write
. Then
is a field known as the extended centroid of
, and the subring
of
is called the central closure of
. One can show that
is a prime ring which is centrally closed, namely it contains its extended centroid. This central closure controls the linear identities of
in the sense that if
with
for all
, then there exists an element
with
and
. Martindale's theorem [a6] asserts that a prime ring
satisfies a non-trivial generalized polynomial identity if and only if
has an idempotent
such that
is a minimal right ideal and
is a division algebra that is finite dimensional over
.
If is a non-commutative free algebra in two variables, then
is a domain but
is not. Thus
is in some sense too large an extension of
. In [a3], it was suggested that for any prime ring
, the set
would define a symmetric version of the Martindale ring of quotients. This was shown to be the case in [a12], where
was characterized as the unique (up to isomorphism) ring extension
of
satisfying:
a) if , then there exist
with
;
b) if and
, then
; and
c) if ,
,
and
for all
,
, then there exists a
with
and
for all
,
.
When is a domain, then so is its symmetric Martindale ring of quotients
. Furthermore, any non-commutative free algebra is symmetrically closed.
An interesting example here is as follows. Let denote the
-vector space of all square matrices of some infinite size, and let
be the subspace which is the direct sum of the scalar matrices and the matrices with only finitely many non-zero entries. Then
is a prime ring,
is the ring of row-finite matrices in
,
is the ring of column-finite matrices, and
is the ring consisting of matrices which are both row and column finite. Thus, in some rough sense,
is the intersection of the left and right Martindale rings of quotients. Other examples of interest can be found in [a2], [a4], [a5], [a8], [a13].
Another important intermediate ring is the normal closure of , defined in [a10] as the product
, where
is the multiplicatively closed set of all units
with
. Then
, and
is the smallest ring extension of
needed to study all group actions on
. Despite its name, the normal closure is not necessarily normally closed. Again, numerous examples of these normal closures have been computed. See, for example, [a7], [a9], [a11].
Finally, as was pointed out in [a1], there is a more general construction which yields analogues of the Martindale ring of quotients for rings which are not necessarily prime. To this end, let be an arbitrary ring (with
) and let
be a non-empty filter of ideals of
. Specifically, it is assumed that:
every ideal is regular, that is, has trivial right and left annihilator in
;
if , then
; and
if and if
with
, then
. Given such a filter, one can again consider all pairs
with
and with
, and use these to construct a ring extension of
which might be denoted by
. For example, if
is a semi-prime ring (cf. Prime ring), then the set
of all regular ideals is such a filter. Here, if
, then the centre
is no longer a field, in general, but it is at least a commutative regular ring (in the sense of von Neumann). Another example of interest occurs when
is a
-prime ring, where
is a fixed group of automorphisms of
. In this case, one can take
to be the set of non-zero
-stable ideals of
, and then the action of
on
extends to an action on
.
References
[a1] | S.A. Amitsur, "On rings of quotients" , Symposia Math. , VIII , Acad. Press (1972) pp. 149–164 |
[a2] | P. Ara, A. del Rio, "A question of Passman on the symmetric ring of quotients" Israel J. Math. , 68 (1989) pp. 348–352 |
[a3] | V.K. Kharchenko, "Generalized identities with automorphisms" Algebra and Logic , 14 (1976) pp. 132–148 Algebra i Logika , 14 (1975) pp. 215–237 |
[a4] | V.K. Kharchenko, "Algebras of invariants of free algebras" Algebra and Logic , 17 (1979) pp. 316–321 Algebra i Logika , 17 (1978) pp. 478–487 |
[a5] | J. Lewin, "The symmetric ring of quotients of a ![]() |
[a6] | W.S. Martindale III, "Prime rings satisfying a generalized polynomial identity" J. Algebra , 12 (1969) pp. 576–584 |
[a7] | W.S. Martindale III, "The normal closure of the coproduct of rings over a division ring" Trans. Amer. Math. Soc. , 293 (1986) pp. 303–317 |
[a8] | W.S. Martindale III, "The symmetric ring of quotients of the coproduct of rings" J. Algebra , 143 (1991) pp. 295–306 |
[a9] | W.S. Martindale III, S. Montgomery, "The normal closure of coproducts of domains" J. Algebra , 82 (1983) pp. 1–17 |
[a10] | S. Montgomery, "Automorphism groups of rings with no nilpotent elements" J. Algebra , 60 (1979) pp. 238–248 |
[a11] | S. Montgomery, "X-inner automorphisms of filtered algebras" Proc. Amer. Math. Soc. , 83 (1981) pp. 263–268 |
[a12] | D.S. Passman, "Computing the symmetric ring of quotients" J. Algebra , 105 (1987) pp. 207–235 |
[a13] | J.D. Rosen, M.P. Rosen, "The Martindale ring of quotients of a skew polynomial ring of automorphism type" Commun. Algebra , 21 (1993) pp. 4051–4063 |
Martindale ring of quotients. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Martindale_ring_of_quotients&oldid=39309