von Neumann Masanori ring

von Neumann Masanori ring (British: von Neumann regular ring) is ring R, and, in mathematics, x ∈ R which there is for any a ∈ R exists, and it is virtually in a = axa [¹]; [²]. The von Neumann Masanori ring is certainly called the flatness ring (absolutely flat ring) to avoid the confusion with Masanori ring and the Masanori local site ring in the commutation ring theory. The von Neumann Masanori ring is because any left module is characterized as the ring that is flatness [³].

I can regard x as "weak inverse element (English version)" (weak inverse) of a. Generally, x is not decided uniquely by a.

The von Neumann Masanori ring was introduced in a study of von Neumann multi dimensions ring and consecutive geometry by the name of "Masanori ring" by von Neumann (1936).

When, in former a of the ring, x becoming a = axa exists; called the origin of von Neumann Masanori [⁴]. Ideal ${\displaystyle {\mathfrak {i}}}$ When is the ring of the non-unit that is は von Neumann Masanori;, in other words ${\displaystyle {\mathfrak {i}}}$ For former a of the の option ${\displaystyle {\mathfrak {i}}}$ When の former x exists, and become a = axa (von Neumann); called the Masanori ideal [⁵].

Table of contents

Example

All bodies (all 可除環) are von Neumann Masanori. I get it for a ≠ 0 with x = a^-1 [⁴]. It is the equivalent that an integral domain is von Neumann Masanori and to be a body.

The different example of the von Neumann Masanori ring is all ingredient baggage n next line ring M_n(K) by the cause of body K. Reversible line U and V exist if I assume r a rank of A ∈ M_n(K) and

${\displaystyle A=U{\begin{pmatrix}I_{r}&0\\0&0\end{pmatrix}}V}$ 

となる (but as for the I_r r next identity matrix). If I put it with X = V^-1U^-1,

${\displaystyle AXA=U{\begin{pmatrix}I_{r}&0\\0&0\end{pmatrix}}{\begin{pmatrix}I_{r}&0\\0&0\end{pmatrix}}V=U{\begin{pmatrix}I_{r}&0\\0&0\end{pmatrix}}V=A}$ 

である. Generally a line ring in the von Neumann Masanori ring is von Neumann Masanori ring again [⁴].

The ring of the affiliated operator (English version) of the limited von Neumann ring is von Neumann Masanori.

Tamaki Bour is the ring that all causes satisfy a2 = a. All Tamaki Bour is von Neumann Masanori.

Fact

The next is the equivalent about ring R.

• R is von Neumann Masanori
• All monomial left ideal is generated by the causes such as a certain one ベキ
• All limited generation left ideal is generated by the causes such as a certain one ベキ
• All monomial left ideal is Naokazu factor of left R-module R
• All limited generation left ideal is Naokazu factor of left R-module R
• All limited generation part modules of projection left R-module P are Naokazu factors of P
• All left R-modules are flat. This is known as R being absolutely flat and a weak dimension of R being 0
• All short perfection line of the left R-module is purely complete (English version) (pure exact)

The thing which changed the left into the right is the equivalent with R being von Neumann Masanori.

Because only former y exists for each former x, and it is in commutation von Neumann Masanori ring in xyx=x and yxy=y, there is the method that is カノニカル choosing "a weak inverse element" of x. The following claims are the equivalent for commutation ring R.

• R is von Neumann Masanori.
• R is Krull dimension 0; a cover about.
• All localization of R in the very large ideal is bodies.
• R is a subring of the prompt shipment of the body closing by operation to take "weak inverse element" (only former y which is xyx=x and yxy=y) of x ∈ R.

In addition, the following is the equivalent, too. For commutation ring A,

• R = A / nil(A) is von Neumann Masanori.
• A spectrum of R is ハウスドルフ (phase a the re-chance).
• The the re-chance aspect to rank as accords with 可設位相 (English version) for Spec(A).

All half simplicity rings are von Neumann Masanori, and the von Neumann Masanori ring of left (or the right) ネーター is semisimple. Jakobsson basis is {0}, and therefore all von Neumann Masanori rings are called the ring ("Jakobsson half simplicity "(Jacobson semi-simple) for the half beginning; is).

I generalize an upper example, and it is an S-module as a ring in M in S, and all partial modules of M shall be Naokazu ingredients of M (called the module semisimple in such module M). Then self-associate same model ring End_S(M) is von Neumann Masanori. Particularly, all half simplicity rings are von Neumann Masanori.

I specialize it with generalization and it

The special type of the von Neumann Masanori ring includes unit Masanori ring (unit regular ring) and strong von Neumann Masanori ring (strongly von Neumann regular ring) and ring (English version) (rank ring) with the number of floors.

When ring R is unit Masanori, for all a ∈ R, unit u ∈ R exists and is that a = aua is managed. All half simplicity rings are unit Masanori, and unit Masanori ring is a Dede quinte limited ring (directly finite ring). As for the normal von Neumann Masanori ring, Dede quinte may not be limited.

When ring R is strong von Neumann Masanori, for all a ∈ R, a certain x ∈ R exists and is that a = aax is managed. This condition is symmetric. The strong von Neumann Masanori ring is unit Masanori. Because all strong von Neumann Masanori rings are expressed by partial prompt shipment (English version) of 可除環, in a sense, as for the strong von Neumann Masanori ring, it is to the thing imitating a property more closely of the commutation von Neumann ring (it is said that I can express it as partial prompt shipment of the commutation body). Of course, for a commutation ring, strong von Neumann Masanori is the equivalent with von Neumann Masanori. Generally, the following is the equivalent for ring R.

• R is strong von Neumann Masanori.
• R von Neumann Masanori and a cover about.
• In R, as for all idempotent elements of von Neumann Masanori and R, central.
• All main left ideals of R are generated by a certain one central idempotent element.

The generalization of the von Neumann Masanori ring includes the following. π-Masanori ring, left / right half heredity ring, a ring non-specific left / right, a semiprimitive ring.

Footnote

1. ^ von Neumann 1960, Definition 2.2.
2. ^ Rotman 2009, p. 159.
3. ^ Rotman 2009, Theorem 4.9 (Harada).
4. ^ ^{a b c} Kaplansky 1972, p. 110
5. ^ Kaplansky 1972, p. 112

References

• Kaplansky, Irving (1972), Fields and rings, Chicago lectures in mathematics (Second ed.), University of Chicago Press, ISBN 0-226-42451-0, Zbl 1001.16500 
• Rotman, Joseph J. (2009). An Introduction to Homological Algebra. Universitext (Second ed.). Springer. ISBN 978-0-387-24527-0. It is https://books.google.com/books?id=P2HV4f8gyCgC. Zbl 1157.18001 

Further reading

• Goodearl, K. R. (1991), von Neumann regular rings (2nd ed.), Malabar, FL: Robert E. Krieger Publishing Co. It is pp. xviii+412, ISBN 0-89464-632-X, MR 1150975, Zbl 0749.16001 Inc 
• L.A. Skornyakov (2001), "Regular ring (in the sense of von Neumann)," it is in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1,556,080,104, http://eom.springer.de/R/r080830.htm 
• von Neumann, John (1936), "On Regular Rings," it is Proc. Nat. Acad. Sci. USA 22 (12): 707–712, doi: 10.1073/pnas.22.12.707, JFM 62.1103.03, PMC 1076849, PMID 16577757, Zbl 0015.38802, http://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=1076849 
• von Neumann, John (1960), Continuous geometries, Princeton University Press, Zbl 0171.28003, https://books.google.com/books?id=onE5HncE-HgC 

Allied item

• Masanori semigroup (English version)
• Weak inverse element (English version)

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