View source for Regular ring (in the sense of von Neumann)

An associative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r0808301.png" /> (usually with unit element) in which the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r0808302.png" /> is solvable for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r0808303.png" /> (cf. [[Associative rings and algebras|Associative rings and algebras]]). The following properties are equivalent: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r0808304.png" /> is a regular ring; b) every principal left ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r0808305.png" /> is generated by an idempotent; c) the principal left ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r0808306.png" /> form a sublattice in the lattice of all left ideals which is a complemented modular lattice; d) every principal left ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r0808307.png" /> has a complement in the lattice of all left ideals; e) all left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r0808308.png" />-modules are flat; and f) the right analogues of the properties b)–e) hold (see [[#References|[3]]], [[#References|[4]]], [[#References|[5]]], [[#References|[8]]], [[#References|[10]]]). In view of e), regular rings are sometimes called absolutely flat. A commutative ring is regular if and only if all simple modules over it are injective (see [[#References|[5]]]). Every finitely-generated left (right) ideal of a regular ring is principal, and is thus distinguished as a direct summand. Every non-divisor of zero in a regular ring is invertible. The [[Jacobson radical|Jacobson radical]] of a regular ring is equal to zero. A ring of matrices over a regular ring is again a regular ring. The class of regular rings is closed under the formation of direct products and quotient rings. An ideal in a regular ring is a regular ring (possibly without unit element). If a regular ring is Noetherian or perfect (left or right), then it is a [[Classical semi-simple ring|classical semi-simple ring]]. Every classical semi-simple ring is regular. Moreover, the endomorphism ring of a vector space over a skew-field is regular (even in the infinite-dimensional case), and so is the quotient ring of the endomorphism ring of any injective left (right) module over any ring by its Jacobson radical (see [[#References|[3]]]). In particular, any left (right) self-injective ring with zero Jacobson radical is regular. The group ring of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r0808309.png" /> over a regular ring is regular if and only if every finitely-generated subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083010.png" /> is finite and the order of every such subgroup is invertible in the original regular ring (see [[#References|[3]]]). The endomorphism rings of all free left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083011.png" />-modules are regular only in the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083012.png" /> is classically semi-simple [[#References|[6]]]. Countably-generated one-sided ideals of regular rings are projective [[#References|[8]]].

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083013.png" /> is a regular ring, then the finitely-generated submodules of the left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083014.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083015.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083016.png" />-tuples over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083017.png" /> form a complemented modular lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083018.png" /> which is a sublattice of the lattice of all submodules of the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083019.png" />. The lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083020.png" /> contains a homogeneous basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083021.png" />, that is, these elements are independent (see [[Modular lattice|Modular lattice]]), their sum is equal to the largest element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083022.png" /> (namely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083023.png" />), and every pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083024.png" /> is perspective, that is, they have a common complement. Conversely, every complemented modular lattice with a homogeneous basis of at least four elements is isomorphic to such a lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083025.png" /> for some regular ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083026.png" />. The lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083027.png" /> is isomorphic to the lattice of principal left ideals of the ring of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083028.png" />-matrices over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083029.png" /> (see [[#References|[4]]], [[#References|[10]]]).

An important special case of a regular ring is a strictly-regular ring, in which, by definition, the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083030.png" /> is always solvable. The following properties of a regular ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083031.png" /> are equivalent: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083032.png" /> is strictly regular; 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083033.png" /> has no non-zero nilpotent elements; 3) all the idempotents of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083034.png" /> are central; 4) every left (or right) ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083035.png" /> is two-sided; 5) the lattice of principal left (right) ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083036.png" /> is distributive; and 6) the multiplicative semi-group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083037.png" /> is an inverse semi-group (see [[#References|[7]]], [[#References|[8]]]).

Another subclass of the class of regular rings is that consisting of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083039.png" />-regular rings, in which, by definition, the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083040.png" /> always has an invertible element as a solution. In the class of regular rings, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083041.png" />-regular rings are characterized by the transitivity of perspectivity in the lattice of finitely-generated submodules of the sum of two copies of the ground ring, and also by the fact that a direct sum can be contracted on a finitely-generated projective module (see [[#References|[4]]], [[#References|[8]]]).

A regular ring is called left continuous if a lattice of principal left ideals is a continuous geometry (cf. [[Orthomodular lattice|Orthomodular lattice]]). A continuous regular ring is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083042.png" />-regular and decomposes into the direct sum of a strictly-regular ring and a [[Self-injective ring|self-injective ring]]. A pseudo-rank function can be defined on a regular ring, and it is an analogue of a measure on a Boolean algebra. It defines a [[Pseudo-metric|pseudo-metric]]. The completion of a regular ring with respect to this metric turns out to be a self-injective regular ring (see [[#References|[8]]]).

Regular rings are a special case of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083044.png" />-regular rings, in which, by definition, for every element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083045.png" /> there are an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083046.png" /> and a positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083047.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083048.png" />.

As two-sided analogues of regular rings, one can consider biregular rings, in which, by definition, every principal two-sided ideal is generated by a central idempotent (cf. [[Central algebra|Central algebra]]). Every two-sided ideal of a biregular ring is an intersection of maximal two-sided ideals. Every biregular ring with a unit is isomorphic to the ring of global sections with compact support of a sheaf of simple rings with a unit over a compact totally disconnected Hausdorff space, and any such ring of global sections is biregular (see [[#References|[2]]]). In the commutative case, the classes of biregular, strictly regular and regular rings coincide, and one must substitute fields for simple rings in the last theorem above.

Baer rings are close to regular rings, and are defined by the condition that every left (or right, which is equivalent when there is a unit element) annihilator is generated by an idempotent. Examples of Baer rings are the endomorphism ring of a vector space over a skew-field and the ring of bounded operators on a Hilbert space. A Baer ring is said to be Abelian if all its idempotents are central, and (Dedekind) finite if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083049.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083050.png" />. An idempotent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083051.png" /> of a Baer ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083052.png" /> is called Abelian (finite) if the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083053.png" /> is Abelian (Dedekind finite). One distinguishes the following classes of Baer rings: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083054.png" />, finite rings containing an Abelian idempotent that does not belong to any proper direct summand; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083055.png" />, Dedekind infinite rings (that is, rings not containing non-zero finite central idempotents) containing an Abelian idempotent that does not belong to any proper direct summand; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083056.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083057.png" />), Dedekind finite rings without non-zero Abelian idempotents, but containing a finite idempotent that does not belong to any proper direct summand; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083058.png" />, Dedekind infinite rings with the condition mentioned in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083059.png" />; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083060.png" />, rings without non-zero finite idempotents. Every Baer ring decomposes in a unique way into a direct sum of rings of these types (see [[#References|[9]]]).

Regular rings were introduced for the coordinization of continuous geometries, biregular rings in connection with the study of functional representations of rings, and Baer (and Rickart) rings in the study of rings of operators.

Non-associative regular rings have also been studied. See also [[*-regular ring|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083061.png" />-regular ring]]; [[Rickart ring|Rickart ring]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,   "Algèbre commutative" , Masson  (1983)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Dauns,   K. Hofmann,   "The representation of biregular rings by sheaves"  ''Math. Z.'' , '''91'''  (1966)  pp. 103–123</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Lambek,   "Lectures on rings and modules" , Blaisdell  (1966)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L.A. Skornyakov,   "Complemented modular lattices and regular rings" , Oliver &amp; Boyd  (1962)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  C. Faith,   "Algebra" , '''1–2''' , Springer  (1973–1976)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  G.M. Tsukerman,   "Ring of endomorphisms of a free module"  ''Sib. Math. J.'' , '''7''' :  5  (1966)  pp. 923–927  ''Sibirsk. Mat. Zh.'' , '''7''' :  5  (1966)  pp. 1161–1167</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  B.M. Shain,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083062.png" />-rings and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083063.png" />-rings"  ''Izv. Vyssh. Uchebn. Mat.'' :  2  (1966)  pp. 111–122  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  K.R. Goodearl,   "Von Neumann regular rings" , Pitman  (1979)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  I. Kaplansky,   "Rings of operators" , Benjamin  (1968)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  J. von Neumann,   "Continuous geometries" , Princeton Univ. Press  (1960)</TD></TR></table>



====Comments====
Quite generally, a rank function, or level function, on a [[Partially ordered set|partially ordered set]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083064.png" /> is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083065.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083066.png" /> (or to another totally ordered set) such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083067.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083068.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083069.png" />. Functions satisfying this property abound in mathematics. For instance, a measure on a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083070.png" />-algebra is (among many other things) a rank function. At this level a rank function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083071.png" /> is simply an order-preserving mapping into a totally ordered set.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083072.png" /> be a partially ordered set with smallest element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083073.png" /> in which all chains between two given elements are finite and satisfying the Jordan–Dedekind chain condition: All maximal chains between two given elements have the same length. Then, by defining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083074.png" /> length of a maximal chain joining zero to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083075.png" /> one obtains a rank function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083076.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083077.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083078.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083079.png" /> immediately precedes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083080.png" />. The value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083081.png" /> is then called the rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083082.png" />.

A rank function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083083.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083084.png" /> on the set of all subsets of a finite set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083085.png" /> is a [[Matroid|matroid]] if, in addition to the order-preserving property, it satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083086.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083087.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083088.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083089.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083090.png" />, [[#References|[a1]]].

In various different contexts, rank functions on ordered sets with additional structure are required to satisfy additional properties.

A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083091.png" /> from a Boolean algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083092.png" /> into the ordinals is called a rank function if it satisfies

i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083093.png" />;

ii) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083094.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083095.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083096.png" /> disjoint and non-zero, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083097.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083098.png" />.

Here, the order relation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r08083099.png" /> is defined by: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830100.png" />; cf. [[#References|[a3]]].

A pointed monoid is a commutative [[Monoid|monoid]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830101.png" /> with an extra designated element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830102.png" />. A Boolean algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830103.png" /> is measured by a pointed monoid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830104.png" /> if there is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830105.png" />, called a measure, such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830106.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830107.png" />:

iii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830108.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830109.png" />;

iv) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830110.png" />;

v) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830111.png" />;

vi) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830112.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830113.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830114.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830115.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830116.png" />.

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830117.png" /> means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830118.png" /> is the disjoint union of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830119.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830120.png" />. The measure isomorphism theorem for Boolean algebras says that countable Boolean algebras measured by the same pointed monoid are isomorphic, [[#References|[a4]]].

A pseudo-rank function on a von Neumann regular ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830121.png" /> is a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830122.png" /> such that

vii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830123.png" />;

viii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830124.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830125.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830126.png" />;

ix) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830127.png" /> for all orthogonal idempotents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830128.png" /> (so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830129.png" />).

It is a rank function if, moreover,

x) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830130.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830131.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830132.png" />.

Given a pseudo-rank function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830133.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830134.png" />, the assignment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830135.png" /> defines a [[Pseudo-metric|pseudo-metric]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830136.png" />, which is a [[Metric|metric]] if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080830/r080830137.png" /> is a rank function; cf. [[#References|[8]]], p. 226ff, for an account of these concepts and their applications.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Whitney,   "On the abstract properties of linear dependence"  ''Amer. J. Math.'' , '''57'''  (1935)  pp. 509–533  ((Reprinted in: Joseph P.S. Kung (ed.), A source book in matroid theory, Birkhäuser, 1986, pp. 55–80))</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Aigner,   "Combinatorial theory" , Springer  (1979)  pp. Chapt. II  (Translated from German)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Koppelberg,   "General theory of Boolean algebras"  J.D. Monk (ed.)  R. Bonnet (ed.) , ''Handbook of Boolean algebras'' , '''1''' , North-Holland  (1989)  pp. 283</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  D. de Myers,   "Lindenbaum–Tarski algebras"  J.D. Monk (ed.)  R. Bonnet (ed.) , ''Handbook of Boolean algebras'' , '''3''' , North-Holland  (1989)  pp. 1167–1195</TD></TR></table>