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

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, 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 -fir" Commun. Algebra , 16 (1988) pp. 1727–1732
[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