WOLFRAM|DEMONSTRATIONS PROJECT

Special Regular Rings with Involution

​
modulus
3
7
11
table for
2
x
+
2
y
solution of equation
2
x
+
2
y
+
2
z
= 0
x
y
2
x
+
2
y
0
0
0
0
1
1
0
2
4
1
0
1
1
1
2
1
2
5
2
0
4
2
1
5
2
2
1
x
y
z
0
0
0
1
2
3
1
2
4
1
3
2
1
3
5
1
4
2
1
4
5
1
5
3
1
5
4
2
1
3
2
1
4
2
3
1
2
3
6
2
4
1
2
4
6
2
6
3
2
6
4
3
1
2
3
1
5
3
2
1
3
2
6
3
5
1
3
5
6
3
6
2
3
6
5
4
1
2
4
1
5
4
2
1
4
2
6
4
5
1
4
5
6
4
6
2
4
6
5
5
1
3
5
1
4
5
3
1
5
3
6
5
4
1
5
4
6
5
6
3
5
6
4
6
2
3
6
2
4
6
3
2
6
3
5
6
4
2
6
4
5
6
5
3
6
5
4
A ring
R
is regular if for every
x∈R
there exists an
y∈R
such that
xyx=x
. Every field is a regular ring. A regular ring with an involution * is called *-regular if
x
*
x
=0
implies
x=0
. The *-regular rings derived from W*-algebras have many special properties. This Demonstration considers three of them.
(A)
a
1
*
a
1
+
a
2
*
a
2
+...+
a
n
*
a
n
=0
implies
a
1
=
a
2
=⋯=
a
n
=0
for any positive integer
n
.
(A')
a
1
*
a
1
+
a
2
*
a
2
=0
implies
a
1
=
a
2
=0
.
(B) For all
a,b∈R
, there exists an element
x∈R
such that
x
*
x
=a
*
a
+b
*
b
.
A implies A', but not conversely.
Let
R
be the field of residues modulo a prime
p
, where
p≡-1(mod4)
. This is *-regular if we take the identity map as involution. In this ring, the equality
x
*
x
+y
*
y
=
2
x
+
2
y
=0
implies
x=y=0
. So A' holds. But the equation
2
x
+
2
y
+
2
z
=0
has nontrivial solutions, as can be seen for
p=3,7,11
. So A does not hold.
If a ring satisfies A' and B, then evidently it satisfies A.