nLab von Neumann regular ring

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Not to be confused with regular local rings in commutative algebra.

Contents

Idea

A von Neumann regular ring is one where all principal left ideals and principal right ideals are generated by idempotents. They were introduced by John von Neumann in 1936 as part of his algebraization of quantum mechanics.

The principal left ideals in a von Neumann regular ring form a sublattice of the lattice of all left ideals which is a complemented modular lattice, and similarly with “left” replaced by “right”. This sets up a connection between von Neumann regular rings and the approach to quantum logic based on lattices: if the elements of a von Neumann regular ring are thought of as observables, the lattice of principal left (or right) ideals can be thought of as consisting of propositions.

Definition

A von Neumann regular ring is a ring AA (not necessarily commutative) such that any of these equivalent conditions hold:

  1. For every element aAa \in A there exists xAx \in A satisfying

    a=axa. a \;=\; a \, x \, a \,.
  2. For every aAa \in A there is an idempotent eAe \in A such that Aa=AeA a = A e. In other words, every principal left ideal of AA is generated by an idempotent.

  3. For every aAa \in A there is an idempotent eAe \in A such that aA=eAa A = e A. In other words, every principal right ideal of AA is generated by an idempotent.

  4. Every principal left ideal of AA is a direct summand of the left AA-module AA.

  5. Every principal right ideal of AA is a direct summand of the right AA-module AA.

  6. Every left AA-module is flat.

  7. Every right AA-module is flat.

To illustrate the use of the rather curious-looking equation a=axaa = a x a, consider this result, which implies that 1 \iff 3.

Lemma

Suppose aAa \in A. If for some xAx \in A we have a=axaa = a x a, then e=axe = a x is idempotent and aA=eAa A = e A. Conversely, if aA=eAa A = e A for some idempotent eAe \in A, then a=axaa = a x a for some xAx \in A, and we can take e=axe = a x.

Proof

For the first implication, suppose for some xAx \in A we have a=axaa = a x a. Then ax=axaxa x = a x a x so e=axe = a x is idempotent, and we just need to show aA=eAa A = e A. Since e=axe = a x we have eAaAe A \subseteq a A. On the other hand, since a=axa=ea a = a x a = e a we have aAeAa A \subseteq e A.

For the converse implication, suppose aA=eAa A = e A for some idempotent ee. This is equivalent to the conjunction of three conditions: eaAe \in a A, aeAa \in e A, and e 2=ee^2 = e. The first condition says e=axe = a x for some xAx \in A. The second condition says a=eya = e y for some yAy \in A, which by the third condition implies ea=eey=ey=ae a = e e y = e y = a, which by the first implies axa=aa x a = a, as desired.

A symmetrical argument, i.e. one applied to the opposite ring, shows that 1 \iff 3. A principal left ideal is a summand of the left AA-module AA if it is generated by an idempotent ee, since then A=AeA(1e)A = Ae \oplus A(1-e), and conversely any summand of the left AA-module AA is generated by an idempotent. It follows that 2 \iff 4, and a symmetrical argument shows that 3 \iff 5.

Examples

References

Von Neumann regular rings were introduced in

See also:

Last revised on December 25, 2024 at 12:40:57. See the history of this page for a list of all contributions to it.