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 $R$ be a prime ring (with $1$) and consider all pairs $( A , f )$, where $A$ is a non-zero ideal of $R$ and where $f : \square _ { R } A \rightarrow \square _ { R } R$ is a left $R$-module mapping. One says that $( A , f )$ and $( A ^ { \prime } , f ^ { \prime } )$ are equivalent if $f$ and $f ^ { \prime }$ agree on their common domain $A \cap A ^ { \prime }$. This is easily seen to yield an equivalence relation, and the set $Q_{\text{l}} ( R )$ of all equivalence classes $[ A , f ]$ is a ring extension of $R$ with arithmetic defined by

\begin{equation*} [ A , f ] + [ B , g ] = [ A \bigcap B , f + g ], \end{equation*}

\begin{equation*} [A,f ] [ B , g ] = [ B A , f g ]. \end{equation*}

Here, $f g$ indicates the mapping $f$ followed by the mapping $g$.

One can show (see [a12]) that the left Martindale ring of quotients $Q_{\text{l}} ( R )$ is characterized as the unique (up to isomorphism) ring extension $Q$ of $R$ satisfying:

1) if $q \in Q$, then there exists a $0 \neq A \lhd R$ with $A q \subseteq R$;

2) if $0 \neq q \in Q$ and $0 \neq I \triangleleft R$, then $Iq \neq 0$; and

3) if $0 \neq A \lhd R$ and $f : \square _ { R } A \rightarrow \square _ { R } R$, then there exists a $q \in Q$ with $a f = a q$ for all $a \in A$. As a consequence, if $R$ is simple, then $Q _ { \text{l} } ( R ) = R$. In any case, $Q_{\text{l}} ( R )$ is certainly a prime ring. The right Martindale ring of quotients $Q _ { r } ( R )$ is defined in an analogous manner and enjoys similar properties.

Again, let $R$ be a prime ring and write $Q = Q _ { \text{l} } ( R )$. Then $C = \mathbf{Z} ( R ) = \mathbf{C} _ { Q } ( R )$ is a field known as the extended centroid of $R$, and the subring $R C$ of $Q$ is called the central closure of $R$. One can show that $R C$ is a prime ring which is centrally closed, namely it contains its extended centroid. This central closure controls the linear identities of $R$ in the sense that if $0 \neq a , b , c , d \in R$ with $axb=cxd$ for all $x \in R$, then there exists an element $0 \neq q \in C$ with $c = a q$ and $d = q ^ { - 1 } b$. Martindale's theorem [a6] asserts that a prime ring $R$ satisfies a non-trivial generalized polynomial identity if and only if $R C$ has an idempotent $e$ such that $e R C$ is a minimal right ideal and $eR Ce$ is a division algebra that is finite dimensional over $C$.

If $R = F \langle x , y \rangle$ is a non-commutative free algebra in two variables, then $R$ is a domain but $Q_{\text{l}} ( R )$ is not. Thus $Q_{\text{l}} ( R )$ is in some sense too large an extension of $R$. In [a3], it was suggested that for any prime ring $R$, the set $Q _ { s } ( R ) = \{ q \in Q_{\text{l} } ( R ) : q B \subseteq R \ \text { for some } \ 0 \neq B \lhd R \}$ would define a symmetric version of the Martindale ring of quotients. This was shown to be the case in [a12], where $Q _ { s } ( R )$ was characterized as the unique (up to isomorphism) ring extension $Q$ of $R$ satisfying:

a) if $q \in Q$, then there exist $0 \neq A , B \lhd R$ with $A q , q B \subseteq R$;

b) if $0 \neq q \in Q$ and $0 \neq I \triangleleft R$, then $I q , q I \neq 0$; and

c) if $0 \neq A , B \lhd R$, $f : \square _ { R } A \rightarrow \square _ { R } R$, $g : B _ { R } \rightarrow R _ { R }$ and $( a f ) b = a ( g b )$ for all $a \in A$, $b \in B$, then there exists a $q \in Q$ with $a f = a q$ and $g b = q b $ for all $a \in A$, $b \in B$.

When $R$ is a domain, then so is its symmetric Martindale ring of quotients $Q _ { s } ( R )$. Furthermore, any non-commutative free algebra is symmetrically closed.

An interesting example here is as follows. Let $\mathcal{M} _ { \infty } ( F )$ denote the $F$-vector space of all square matrices of some infinite size, and let $R$ be the subspace which is the direct sum of the scalar matrices and the matrices with only finitely many non-zero entries. Then $R$ is a prime ring, $Q_{\text{l}} ( R )$ is the ring of row-finite matrices in $\mathcal{M} _ { \infty } ( F )$, $Q _ { r } ( R )$ is the ring of column-finite matrices, and $Q _ { s } ( R )$ is the ring consisting of matrices which are both row and column finite. Thus, in some rough sense, $Q _ { s } ( R )$ 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 $R$, defined in [a10] as the product $R N$, where $N$ is the multiplicatively closed set of all units $u \in Q _ { \text{l} } ( R )$ with $u ^ { - 1 } R u = R$. Then $R C \subseteq R N \subseteq Q _ { s } ( R )$, and $R N$ is the smallest ring extension of $R$ needed to study all group actions on $R$. 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 $R$ be an arbitrary ring (with $1$) and let $\mathcal{F}$ be a non-empty filter of ideals of $R$. Specifically, it is assumed that:

every ideal $A \in \mathcal{F}$ is regular, that is, has trivial right and left annihilator in $R$;

if $A , B \in \cal{F}$, then $A B \in \mathcal{F}$; and

if $A \in \mathcal{F}$ and if $B \triangleleft R$ with $A \subseteq B$, then $B \in \mathcal{F}$. Given such a filter, one can again consider all pairs $( A , f )$ with $A \in \mathcal{F}$ and with $f : \square _ { R } A \rightarrow \square _ { R } R$, and use these to construct a ring extension of $R$ which might be denoted by $Q _ { \mathcal{F} } ( R )$. For example, if $R$ is a semi-prime ring, then the set $\mathcal{F}$ of all regular ideals is such a filter. Here, if $Q = Q _ { \mathcal{F} } ( R )$, then the centre $C = \mathbf{Z} ( Q )$ 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 $R$ is a $G$-prime ring, where $G$ is a fixed group of automorphisms of $R$. In this case, one can take $\mathcal{F}$ to be the set of non-zero $G$-stable ideals of $R$, and then the action of $G$ on $R$ extends to an action on $Q _ { \mathcal{F} } ( R )$.

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 $2$-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