Regular ring (in the sense of von Neumann)

An associative ring $ R $( usually with unit element) in which the equation $ axa = a $ is solvable for any $ a $( cf. Associative rings and algebras). The following properties are equivalent: a) $ R $ is a regular ring; b) every principal left ideal of $ R $ is generated by an idempotent; c) the principal left ideals of $ R $ form a sublattice in the lattice of all left ideals which is a complemented modular lattice; d) every principal left ideal of $ R $ has a complement in the lattice of all left ideals; e) all left $ R $- modules are flat; and f) the right analogues of the properties b)–e) hold (see [3], [4], [5], [8], [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 [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 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. 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 [3]). In particular, any left (right) self-injective ring with zero Jacobson radical is regular. The group ring of a group $ G $ over a regular ring is regular if and only if every finitely-generated subgroup of $ G $ is finite and the order of every such subgroup is invertible in the original regular ring (see [3]). The endomorphism rings of all free left $ R $- modules are regular only in the case when $ R $ is classically semi-simple [6]. Countably-generated one-sided ideals of regular rings are projective [8].

If $ R $ is a regular ring, then the finitely-generated submodules of the left $ R $- module $ R ^ {n} $ of $ n $- tuples over $ R $ form a complemented modular lattice $ L $ which is a sublattice of the lattice of all submodules of the module $ R ^ {n} $. The lattice $ L $ contains a homogeneous basis $ a _ {1} \dots a _ {n} $, that is, these elements are independent (see Modular lattice), their sum is equal to the largest element of $ L $( namely $ R ^ {n} $), and every pair $ a _ {i} , a _ {j} $ 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 $ L $ for some regular ring $ R $. The lattice $ L $ is isomorphic to the lattice of principal left ideals of the ring of all $ ( n \times n ) $- matrices over $ R $( see [4], [10]).

An important special case of a regular ring is a strictly-regular ring, in which, by definition, the equation $ a ^ {2} x = a $ is always solvable. The following properties of a regular ring $ R $ are equivalent: 1) $ R $ is strictly regular; 2) $ R $ has no non-zero nilpotent elements; 3) all the idempotents of $ R $ are central; 4) every left (or right) ideal of $ R $ is two-sided; 5) the lattice of principal left (right) ideals of $ R $ is distributive; and 6) the multiplicative semi-group of $ R $ is an inverse semi-group (see [7], [8]).

Another subclass of the class of regular rings is that consisting of $ u $- regular rings, in which, by definition, the equation $ axa = a $ always has an invertible element as a solution. In the class of regular rings, $ u $- 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 [4], [8]).

A regular ring is called left continuous if a lattice of principal left ideals is a continuous geometry (cf. Orthomodular lattice). A continuous regular ring is $ u $- regular and decomposes into the direct sum of a strictly-regular ring and a 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. The completion of a regular ring with respect to this metric turns out to be a self-injective regular ring (see [8]).

Regular rings are a special case of $ \pi $- regular rings, in which, by definition, for every element $ a $ there are an element $ x $ and a positive integer $ n $ such that $ a ^ {n} x a ^ {n} = a ^ {n} $.

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). 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 [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 $ xy = 1 $ implies $ yx = 1 $. An idempotent $ e $ of a Baer ring $ R $ is called Abelian (finite) if the ring $ e R e $ is Abelian (Dedekind finite). One distinguishes the following classes of Baer rings: $ I _ {\textrm{ fin } } $, finite rings containing an Abelian idempotent that does not belong to any proper direct summand; $ I _ {\textrm{ inf } } $, 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; $ II _ {\textrm{ fin } } $( or $ II _ {1} $), Dedekind finite rings without non-zero Abelian idempotents, but containing a finite idempotent that does not belong to any proper direct summand; $ II _ {\textrm{ inf } } $, Dedekind infinite rings with the condition mentioned in $ II _ {\textrm{ fin } } $; and $ III $, rings without non-zero finite idempotents. Every Baer ring decomposes in a unique way into a direct sum of rings of these types (see [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; Rickart ring.

References

[1]	N. Bourbaki, "Algèbre commutative" , Masson (1983)
[2]	J. Dauns, K. Hofmann, "The representation of biregular rings by sheaves" Math. Z. , 91 (1966) pp. 103–123
[3]	J. Lambek, "Lectures on rings and modules" , Blaisdell (1966)
[4]	L.A. Skornyakov, "Complemented modular lattices and regular rings" , Oliver & Boyd (1962) (Translated from Russian)
[5]	C. Faith, "Algebra" , 1–2 , Springer (1973–1976)
[6]	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
[7]	B.M. Shain, "-rings and -rings" Izv. Vyssh. Uchebn. Mat. : 2 (1966) pp. 111–122 (In Russian)
[8]	K.R. Goodearl, "Von Neumann regular rings" , Pitman (1979)
[9]	I. Kaplansky, "Rings of operators" , Benjamin (1968)
[10]	J. von Neumann, "Continuous geometries" , Princeton Univ. Press (1960)

Comments

Quite generally, a rank function, or level function, on a partially ordered set $ P $ is a function $ \varphi : P \rightarrow \mathbf R $ or $ \mathbf N $( or to another totally ordered set) such that $ \varphi ( a) \leq \varphi ( b) $ if $ a < b $ in $ P $. Functions satisfying this property abound in mathematics. For instance, a measure on a $ \sigma $- algebra is (among many other things) a rank function. At this level a rank function on $ P $ is simply an order-preserving mapping into a totally ordered set.

Let $ P $ be a partially ordered set with smallest element $ 0 $ 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 $ \rho ( a) = $ length of a maximal chain joining zero to $ a $ one obtains a rank function $ \rho : P \rightarrow \mathbf N \cup \{ 0 \} $ satisfying $ \rho ( 0) = 0 $ and $ \rho ( b) = \rho ( a) + 1 $ if $ a $ immediately precedes $ b $. The value $ \rho ( a) $ is then called the rank of $ a $.

A rank function $ \rho $ to $ \mathbf N \cup \{ 0 \} $ on the set of all subsets of a finite set $ M $ is a matroid if, in addition to the order-preserving property, it satisfies $ \rho ( \emptyset ) = 0 $, $ \rho ( N \cup \{ e \} ) \leq \rho ( N) + 1 $, $ \rho ( N \cup \{ e _ {1} , e _ {2} \} ) = \rho ( N) $ if $ \{ e _ {1} , e _ {2} \} \cap N = \emptyset $ and $ \rho ( N) = \rho ( N \cup \{ e _ {1} \} ) = \rho ( N \cup \{ e _ {2} \} ) $, [a1].

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

A mapping $ r $ from a Boolean algebra $ A $ into the ordinals is called a rank function if it satisfies

i) $ b \leq a \Rightarrow r( b) \leq r( a) $;

ii) if $ a = b+ c $ with $ b $ and $ c $ disjoint and non-zero, then $ r( b) < r ( a) $ or $ r( c) < r( a) $.

Here, the order relation on $ A $ is defined by: $ b \leq a \iff ba = b $; cf. [a3].

A pointed monoid is a commutative monoid $ ( M, + , 0) $ with an extra designated element $ m _ {0} $. A Boolean algebra $ A $ is measured by a pointed monoid $ ( M , + , 0 , m _ {0} ) $ if there is a function $ \mu : A \rightarrow M $, called a measure, such that for all $ a, b \in A $, $ m, n \in M $:

iii) $ \mu ( a) = 0 $ if and only if $ a = 0 $;

iv) $ \mu ( 1) = m _ {0} $;

v) $ \mu ( a \dot{+} b) = \mu ( a) + \mu ( b) $;

vi) if $ \mu ( a) = m+ n $, then $ a = b \dot{+} c $ for some $ b, c \in A $ with $ \mu ( b) = m $, $ \mu ( c) = n $.

Here $ a = b \dot{+} c $ means that $ a $ is the disjoint union of $ b $ and $ c $. The measure isomorphism theorem for Boolean algebras says that countable Boolean algebras measured by the same pointed monoid are isomorphic, [a4].

A pseudo-rank function on a von Neumann regular ring $ R $ is a mapping $ N : R \rightarrow [ 0, 1] $ such that

vii) $ N( 1) = 1 $;

viii) $ N( xy) \leq N( x) $, $ N( xy) \leq N( y) $ for all $ x, y \in R $;

ix) $ N( e+ f ) = N( e) + N( f ) $ for all orthogonal idempotents $ e, f \in R $( so that $ N( 0) = 0 $).

It is a rank function if, moreover,

x) $ N( x) > 0 $ for all $ x \neq 0 $ in $ R $.

Given a pseudo-rank function $ N $ on $ R $, the assignment $ \delta ( x, y) = N( x- y) $ defines a pseudo-metric on $ R $, which is a metric if $ N $ is a rank function; cf. [8], p. 226ff, for an account of these concepts and their applications.

References

[a1]	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))
[a2]	M. Aigner, "Combinatorial theory" , Springer (1979) pp. Chapt. II (Translated from German)
[a3]	S. Koppelberg, "General theory of Boolean algebras" J.D. Monk (ed.) R. Bonnet (ed.) , Handbook of Boolean algebras , 1 , North-Holland (1989) pp. 283
[a4]	D. de Myers, "Lindenbaum–Tarski algebras" J.D. Monk (ed.) R. Bonnet (ed.) , Handbook of Boolean algebras , 3 , North-Holland (1989) pp. 1167–1195