Special Regular Rings with Involution
Special Regular Rings with Involution
A ring is regular if for every there exists an such that . Every field is a regular ring. A regular ring with an involution * is called *-regular if implies . The *-regular rings derived from W*-algebras have many special properties. This Demonstration considers three of them.
R
x∈R
y∈R
xyx=x
x=0
*
x
x=0
(A) ++...+=0 implies ==⋯==0 for any positive integer .
a
1
*
a
1
a
2
*
a
2
a
n
*
a
n
a
1
a
2
a
n
n
(A') +=0 implies ==0.
a
1
*
a
1
a
2
*
a
2
a
1
a
2
(B) For all , there exists an element such that .
a,b∈R
x∈R
x=a+b
*
x
*
a
*
b
A implies A', but not conversely.
Let be the field of residues modulo a prime , where . This is *-regular if we take the identity map as involution. In this ring, the equality implies . So A' holds. But the equation ++=0 has nontrivial solutions, as can be seen for . So A does not hold.
R
p
p≡-1(mod4)
x+y=+=0
*
x
*
y
2
x
2
y
x=y=0
2
x
2
y
2
z
p=3,7,11
If a ring satisfies A' and B, then evidently it satisfies A.