View source for Martindale ring of quotients

This ring of quotients was introduced in [[#References|[a6]]] as a tool to study prime rings satisfying a generalized polynomial identity. Specifically, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m1201201.png" /> be a [[Prime ring|prime ring]] (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m1201202.png" />) and consider all pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m1201203.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m1201204.png" /> is a non-zero ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m1201205.png" /> and where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m1201206.png" /> is a left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m1201207.png" />-module mapping. One says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m1201208.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m1201209.png" /> are equivalent if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012012.png" /> agree on their common domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012013.png" />. This is easily seen to yield an equivalence relation, and the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012014.png" /> of all equivalence classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012015.png" /> is a ring extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012016.png" /> with arithmetic defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012017.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012018.png" /></td> </tr></table>

Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012019.png" /> indicates the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012020.png" /> followed by the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012021.png" />.

One can show (see [[#References|[a12]]]) that the left Martindale ring of quotients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012022.png" /> is characterized as the unique (up to isomorphism) ring extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012023.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012024.png" /> satisfying:

1) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012025.png" />, then there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012026.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012027.png" />;

2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012029.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012030.png" />; and

3) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012032.png" />, then there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012033.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012034.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012035.png" />. As a consequence, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012036.png" /> is simple, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012037.png" />. In any case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012038.png" /> is certainly a prime ring. The right Martindale ring of quotients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012039.png" /> is defined in an analogous manner and enjoys similar properties.

Again, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012040.png" /> be a prime ring and write <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012041.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012042.png" /> is a field known as the extended centroid of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012043.png" />, and the subring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012044.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012045.png" /> is called the central closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012046.png" />. One can show that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012047.png" /> is a prime ring which is centrally closed, namely it contains its extended centroid. This central closure controls the linear identities of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012048.png" /> in the sense that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012049.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012050.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012051.png" />, then there exists an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012052.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012054.png" />. Martindale's theorem [[#References|[a6]]] asserts that a prime ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012055.png" /> satisfies a non-trivial generalized polynomial identity if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012056.png" /> has an idempotent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012057.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012058.png" /> is a minimal right [[Ideal|ideal]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012059.png" /> is a [[Division algebra|division algebra]] that is finite dimensional over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012060.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012061.png" /> is a non-commutative [[Free algebra|free algebra]] in two variables, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012062.png" /> is a domain but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012063.png" /> is not. Thus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012064.png" /> is in some sense too large an extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012065.png" />. In [[#References|[a3]]], it was suggested that for any prime ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012066.png" />, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012067.png" /> would define a symmetric version of the Martindale ring of quotients. This was shown to be the case in [[#References|[a12]]], where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012068.png" /> was characterized as the unique (up to isomorphism) ring extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012069.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012070.png" /> satisfying:

a) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012071.png" />, then there exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012072.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012073.png" />;

b) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012075.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012076.png" />; and

c) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012078.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012079.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012080.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012081.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012082.png" />, then there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012083.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012084.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012085.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012086.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012087.png" />.

When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012088.png" /> is a domain, then so is its symmetric Martindale ring of quotients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012089.png" />. Furthermore, any non-commutative free algebra is symmetrically closed.

An interesting example here is as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012090.png" /> denote the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012091.png" />-vector space of all square matrices of some infinite size, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012092.png" /> be the subspace which is the direct sum of the scalar matrices and the matrices with only finitely many non-zero entries. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012093.png" /> is a prime ring, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012094.png" /> is the ring of row-finite matrices in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012095.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012096.png" /> is the ring of column-finite matrices, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012097.png" /> is the ring consisting of matrices which are both row and column finite. Thus, in some rough sense, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012098.png" /> is the intersection of the left and right Martindale rings of quotients. Other examples of interest can be found in [[#References|[a2]]], [[#References|[a4]]], [[#References|[a5]]], [[#References|[a8]]], [[#References|[a13]]].

Another important intermediate ring is the normal closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m12012099.png" />, defined in [[#References|[a10]]] as the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120100.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120101.png" /> is the multiplicatively closed set of all units <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120102.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120103.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120104.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120105.png" /> is the smallest ring extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120106.png" /> needed to study all group actions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120107.png" />. Despite its name, the normal closure is not necessarily normally closed. Again, numerous examples of these normal closures have been computed. See, for example, [[#References|[a7]]], [[#References|[a9]]], [[#References|[a11]]].

Finally, as was pointed out in [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120108.png" /> be an arbitrary ring (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120109.png" />) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120110.png" /> be a non-empty [[Filter|filter]] of ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120111.png" />. Specifically, it is assumed that:

every ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120112.png" /> is regular, that is, has trivial right and left annihilator in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120113.png" />;

if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120114.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120115.png" />; and

if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120116.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120117.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120118.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120119.png" />. Given such a filter, one can again consider all pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120120.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120121.png" /> and with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120122.png" />, and use these to construct a ring extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120123.png" /> which might be denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120124.png" />. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120125.png" /> is a semi-prime ring (cf. [[Prime ring|Prime ring]]), then the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120126.png" /> of all regular ideals is such a filter. Here, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120127.png" />, then the centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120128.png" /> is no longer a [[Field|field]], in general, but it is at least a commutative [[Regular ring (in the sense of von Neumann)|regular ring (in the sense of von Neumann)]]. Another example of interest occurs when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120129.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120131.png" />-prime ring, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120132.png" /> is a fixed group of automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120133.png" />. In this case, one can take <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120134.png" /> to be the set of non-zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120135.png" />-stable ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120136.png" />, and then the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120137.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120138.png" /> extends to an action on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120120/m120120139.png" />.

====References====
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