Not to be confused with regular local rings in commutative algebra.
symmetric monoidal (∞,1)-category of spectra
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.
A von Neumann regular ring is a ring (not necessarily commutative) such that any of these equivalent conditions hold:
For every element there exists satisfying
For every there is an idempotent such that . In other words, every principal left ideal of is generated by an idempotent.
For every there is an idempotent such that . In other words, every principal right ideal of is generated by an idempotent.
Every principal left ideal of is a direct summand of the left -module .
Every principal right ideal of is a direct summand of the right -module .
Every left -module is flat.
Every right -module is flat.
To illustrate the use of the rather curious-looking equation , consider this result, which implies that 1 3.
Suppose . If for some we have , then is idempotent and . Conversely, if for some idempotent , then for some , and we can take .
For the first implication, suppose for some we have . Then so is idempotent, and we just need to show . Since we have . On the other hand, since we have .
For the converse implication, suppose for some idempotent . This is equivalent to the conjunction of three conditions: , , and . The first condition says for some . The second condition says for some , which by the third condition implies , which by the first implies , as desired.
A symmetrical argument, i.e. one applied to the opposite ring, shows that 1 3. A principal left ideal is a summand of the left -module if it is generated by an idempotent , since then , and conversely any summand of the left -module is generated by an idempotent. It follows that 2 4, and a symmetrical argument shows that 3 5.
Every field is a von Neumann regular ring.
Every semisimple ring is von Neumann regular.
For a vector space over a skew-field , its endomorphism ring is von Neumann regular.
The property of being a von Neumann regular ring is invariant under Morita equivalence, so if is a von Neumann regular ring so is the matrix algebra .
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.