On division rings with general involution
In this work we consider division rings with general involution. Properties of such division rings are investigated. General valuation of such division rings is introduced. We extend to this general notion some result on the extension of valuations.
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irk-123456789-1573492019-06-21T01:29:49Z On division rings with general involution Idris, I.M. In this work we consider division rings with general involution. Properties of such division rings are investigated. General valuation of such division rings is introduced. We extend to this general notion some result on the extension of valuations. 2007 Article On division rings with general involution / I.M. Idris // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 1. — С. 40–48. — Бібліогр.: 4 назв. — англ. 1726-3255 http://dspace.nbuv.gov.ua/handle/123456789/157349 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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In this work we consider division rings with general involution. Properties of such division rings are investigated.
General valuation of such division rings is introduced. We extend
to this general notion some result on the extension of valuations. |
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Idris, I.M. On division rings with general involution Algebra and Discrete Mathematics |
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Idris, I.M. |
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Idris, I.M. |
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On division rings with general involution |
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On division rings with general involution |
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On division rings with general involution |
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On division rings with general involution |
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On division rings with general involution |
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on division rings with general involution |
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Інститут прикладної математики і механіки НАН України |
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2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/157349 |
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On division rings with general involution / I.M. Idris // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 1. — С. 40–48. — Бібліогр.: 4 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT idrisim ondivisionringswithgeneralinvolution |
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2025-07-14T09:47:38Z |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 1. (2007). pp. 40 – 48
c© Journal “Algebra and Discrete Mathematics”
On division rings with general involution
Ismail M. Idris
Communicated by D. Simson
Abstract. In this work we consider division rings with gen-
eral involution. Properties of such division rings are investigated.
General valuation of such division rings is introduced. We extend
to this general notion some result on the extension of valuations.
1. Introduction
In this paper, we study a generalization of the notion of an involution,
which we will refer to as an ε-involution. An ε-involution on a division
ring D is an anti-automorphism ∗ whose square is a conjugation by an
element ε ∈ D such that εε∗ = ε∗ε = 1. In Section 2, we establish
elementary properties and basic facts about the considered division rings.
We show for instance that, if all the symmetric elements in a division ring
D with ε-involution are central, then D is a commutative field.
Valuations of division rings with ε-involution are introduced in Sec-
tion 3. In [1], a necessary and sufficient condition is given for extending
an abelian valuation from a division ring to an over division ring. We
solve here a ∗-version of this problem, where valuations are replaced with
ε-valuations.
2. Properties of division rings with ε-involution
Let D be a division ring with centre Z(D).
Definition 2.1. An anti-automorphism x 7→ x∗ of the division ring D
is called an ε-involution if there is ε ∈ D such that for each x ∈ D,
(x∗)∗ = ε−1xε; where εε∗ = ε∗ε = 1.
Key words and phrases: division rings; involution; valuation.
I. M. Idris 41
The general idea is to consider an anti-automorphism ∗ : D 7→ D on
a division ring D such that (x∗)∗ = ε−1xε for some ε in D (i.e. up to an
inner automorphism it has order 2). Indeed, the map sending x to εx∗ is
a map on D of order 2 (although in general not an anti-automorphism).
As a first observation, when ε = 1, then ∗ is an involution in the usual
sense. More generally, if ε is in the centre Z(D), again ∗ is an involution.
Definition 2.2. Let D be a division ring with ε-involution ∗. An element
x ∈ D is called successively a symmetric element if εx∗ = x, a skew-
symmetric element if εx∗ = −x.
Let S, K denote the additive subgroups of the division ring D formed
by the symmetric elements and the skew-symmetric elements respectively.
In symbols:
S = {x ∈ D|εx∗ = x}; K = {x ∈ D|εx∗ = −x}.
Let T denote the subgroup of traces: T = {x + εx∗|x ∈ D}.
Examples 2.3.
1. For D = C, the complex field, complex conjugation is an ε-
involution for every complex number ε of norm 1.
2. Let H = {i, j|i2 = −1, j2 = −1, ij = −ji} stand for the classical
real quateraionic algebra {a0 + a1i + a2j + a3k|k = ij, ai ∈ R}. Take for
∗ the mapping defined by
(a0 + a1i + a2j + a3k)∗ = a0 − a1i − a2j + a3k (ai ∈ R).
It is easy to check that ∗ is an anti-autmorphism of H such that x∗∗ =
ε−1xε for every x ∈ H, where ε = i. Thus ∗ is an ε-involution. Here the
additive subgroup of symmetric elements is the 1-dimensional subspace
R(1 + i), the additive subgroup of skew symmetric elements is R(i− 1) +
R(j − k) + R(k − j).
3. Let H stand for the classical real quateraionic algebra given in 2.
Take for ∗ the conjugation mapping defined by
(a0 + a1i + a2j + a3k)∗ = a0 − a1i − a2j − a3k (ai ∈ R).
Then, ∗ is an ε-involution with ε = 1. In fact, ∗ is the unique 1-
involution of H such that S ⊃ R, T ⊂ R, and xx∗ ∈ R for all x ∈ R.
We note that H has many 1-involutions with S ⊃ R, in fact any map
x 7→ ux∗u−1, u a unit of H with u∗ = ±u, is an involution. Conversely,
any involution ♯ of H with S = R can be expressed as x♯ = ux∗u−1, for
u∗ = ±u.
42 On division rings with general involution
Remark 2.4. If char(D) 6= 2, then
(1) For each x ∈ D, x = s + k where s is a symmetric element and k is
a skew-symmetric element.
(2) T = S.
(3) S and K are preserved under the 2-sided translation a 7→ xax∗, x ∈
D. If char(D) = 2, we still may have T = S, e.g. if there exists an
element z in the center of D with z∗ 6= z.
Proof. (1) Since char(D) 6= 2 it follows that x = 1
2
(x+x∗)+ 1
2
(x−x∗) =
s + k as desired.
(2) Clearly T ⊂ S. Since each symmetric element can be written in
the form s = (1
2
)s + ε((1
2
)s)∗, it follows that T = S.
(3) If a is a symmetric element, then ε(xax∗)∗ = εε−1xε.a∗.x∗ = xax∗,
so that xax∗ is a symmetric element. Similarly one can prove the result
for a skew-symmetric element a.
We note that any ε-involution ∗ with ε 6= 1 is such that ε is not
symmetric. In fact, if ε is symmetric, then εε∗ = ε, so that ε = 1, a
contradiction. We also note that if k is a skew symmetric element and s
is a symmetric element of D, then the following are symmetric elements:
k(1 + ε)k∗ = kk∗ − k2; s(1 + ε)s∗ = ss∗ + s2 . For a, b ∈ D, put
(a, b) = ab − ba.
Lemma 2.5. Let D be a division ring with ε-involution ∗ such that
char(D) 6= 2, and suppose that sk− ks = 0 for all symmetric s and skew
symmetric k in D. Then for all s ∈ S and, x ∈ D, follows (s, (s, x)) = 0.
Proof. Firstly we claim that ε ∈ Z. Since 1 + ε is a symmetric element,
it follows that 1 + ε commutes with each skew symmetric element in D,
as does ε. Also, since 1 − ε is a skew symmetric element of D, it follows
that 1 − ε commutes with each symmetric element, as does ε. Thus ε
commutes with all symmetrics and skew symmetrics in D. In view of
Remark 2.4, ε is in the centre of D.
Now, let s and s′ be two arbitrary symmetric elements of D. It
is easy to check that (1 + ε∗)(s, s′) is a skew symmetric element; and
consequently, this element commutes with s. Since ε ∈ Z(D), it follows
that ε∗ ∈ Z(D) so that 1 + ε∗ ∈ Z(D). Hence (s, (s, s′)) = 0. Since we
also have (s, (s, k)) = 0 for all skew symmetric elements k ∈ D, it follows
that (s, (s, x)) = 0 for all x ∈ D.
Lemma 2.6 ([2]). Let D be a division ring with char(D) 6= 2. If
(a, (a, x)) = 0 for all x ∈ D, then a is in the centre of D.
I. M. Idris 43
The following corollary is an immediate consequence of the above two
lemmas.
Corollary 2.7. If D is a division ring with ε-involution ∗ such that
sk − ks = 0 for all symmetrics s and skew symmetrics k in D, then all
the symmetric elements of D are central.
Theorem 2.8. Let D be a division ring with ε-involution ∗, where ε 6= 1.
Suppose that all the symmetric elements in D are central. Then D is a
commutative field.
Proof. Since 1 + ε ∈ S ⊂ Z(D), it follows that ε ∈ Z(D). Two cases to
be considered:
(i) If ε 6= −1: Let k0 = 1−ε
1+ε
. Then k0 ∈ Z(D) and ε = 1−k0
1+k0
. Also,
(k0)
∗ = −k0. Now, we claim that each skew symmetric element σ is
central. Because, ε(k0σ)∗ = εσ∗(k0)
∗ = σk0 = k0σ (where k0 ∈ Z(D))
, it follows that k0σ is a symmetric element. Then k0σ ∈ Z(D). Since
k0 ∈ Z(D), it follows that σ ∈ Z(D).
(ii) If ε = −1: In this case s is a symmetric element if and only if
s∗ = −s, and k is a skew symmetric element if and only if k∗ = k. Then
each s = −s∗ is central. For a symmetric element s and a skew symmetric
element k in D one has (sk)∗ = k∗s∗ = −ks = −(sk). Thus sk ∈ Z(D)
giving k ∈ Z(D).
Remark 2.9. Let D be a division ring with ε-involution ∗. If p is
any non-zero element in D, then the formula x♯ = p−1xp defines an
ε1-involution ♯ of D, where ε1 ∈ D can be found.
Proof. Clearly ♯ is an antiautomorphism. Put ε1 = ε(p−1)∗p. It is to be
shown that ♯ is an ε1-invo1ution. For,
x♯♯ = p−1(p−1x∗p)∗p
= p−1p∗ε−1x(p−1)∗p
= ε−1
1
xε1.
One also has ε1ε
♯
1
= ε(p−1)∗pp−1p∗ε−1 = 1. Thus ♯ is an ε1-involution.
By analogy with the usual notion of co-gradient involution, refer to
♯ appearing above to be a co-gradient ε-invo1ution ( ε need not be the
same). It is appropriate to observe that while in the classical case one
requires p to be a symmetric element, here we will dispense with this
requirement. To find a co-gradient involution ♯, all that is needed is that
ε1 = ε(p−1)∗p = 1, meaning that p−1 is a symmetric element relative to
the initial ε-invo1ution.
44 On division rings with general involution
Theorem 2.10. If D is a division ring, then D admits ε-involution if
and only if it admits a usual involution.
Proof. If x∗ = x for each x ∈ D, then evidently D is commutative, in
which case there is nothing more to prove. If, on the other hand, there
is x 6= x∗, we contend that D contains a symmetric element a. For, put
a = x + εx∗. If a 6= 0, then we are done. If, on the other hand, a = 0
then εx∗ = −x, so that b = xx∗ − x∗ is a non-zero symmetric element.
Now, take p to be a−1. Define ♯ by:
a♯ = p−1a∗p(a ∈ D).
Thus ♯ is an involution. Note that if all the symmetric elements are
central, then by Theorem 2.8, D is commutative.
For the converse, take ♯ to be a usual involution of D. Define ∗ of D
by:
x∗ = b−1x♯b (for some b ∈ D, b 6= ±b♯).
Clearly ∗ is an anti-automorphism of D, and x∗∗ = ε−1xε when ε =
(b−1)♯b. Also εε∗ = (b−1)♯bb−1ε♯b = (b−1)♯b♯b−1b = 1, as desired.
Example 2.11. Let D = H the classical real quatrenionic algebra. The
usual involution of D is defined by
(a0 + a1i + a2j + a3k)♯ = a0 − a1i − a2j − a3k.
With reference to the same ε-involution ∗ considered in Example 2.3(2),
(a0 + a1i + a2j + a3k)∗ = a0 − a1i − a2j + a3k, one can say that ∗ is a
co-gradient ε-involution. For, one can check that x∗ = p−1x♯p(x ∈ D),
where p = 1 + i. Changing p = 1 + i to p = 1 + j or p = 1 + k, this gives
two additional ε-involutions of D.
In [3], W. Scharlau, making use of Hilbert’s Theorem, established
that if D is a finite dimensional central simple algebra with an anti-
automorphism θ such that θ2(x) = b−1xb(x ∈ D), where b is such that
θ(b)b = bθ(b) = aθ(a) for some central element a, then D possesses an in-
volution. From Scharlau’s arguments we may restrict our attention to the
case where D is a division ring. In this case the given anti-automorphism
θ is, in fact, an ε-involution. For, if ε = a−1b, then
ε.θ(ε) = (a−1b)θ(a−1)
= a−1bθ(b)θ(a−1) = 1.
Also, ε−1xε = a(b−1xb)a−1 = a(θ2(x))a−1 = θ2(x).
I. M. Idris 45
Therefore, Scharlau’s criterion is simply Theorem 2.10. We note that
in Theorem 2.10, we do not use Hilbert’s Theorem.
The last topic of this section is about the symmetrics and the skew-
symmetrics of (D, ∗) and (D, ♯), where ♯ is a co-gradient ε1-involution of
D(ε1 = ε(p−1)∗p as in Remark 2.9 above).
Theorem 2.12. Let S1 and K1 be the additive subgroups of (D, ♯) of
symmetric elements and skew symmetric elements respectively. Then
S1 = Sp; and K1 = Kp.
Proof. Let s1 ∈ S1. Then ε1s
♯
1
= s1 so that
ε(p∗)−1bb−1s∗1b = s1,
ε(p∗)−1s∗1b = s1.
Hence ε(s1b
−1)∗ = (s1b
−1), and so, s1b
−1 is a symmetric element with
respect to ∗. Thus, S1b
−1 ⊆ S that is S1 ⊆ Sb.
Conversely, if εs∗ = s, then
ε1(sb)
♯ = ε1b
♯s♯
= εs∗b
= sb.
Hence, sb ∈ S1, so that Sb ⊆ S1. Similarly, one can show that K1 =
Kb.
3. ε-Valuations of division rings with ε-involution
Let D be a division ring with ε-involution ∗ and let D• be the multiplica-
tive group of nonzero elements of D . By ε-valuation we mean a mapping
ω : D → G, where G is a linearly ordered additive group with positive
infinity adjoined, such that
1. ω(xy) = ω(x) + ω(y)(x, y ∈ D);
2. ω(x + y) ≥ min{ω(x), ω(y)}(x + y 6= 0);
3. ω maps D onto G; and
4. ω(εx∗) = ω(x).
We first remark that any ε-valuation ω : D → G is abelian, that is,
its value group G is an abelian group, because
ω(x)+ω(y) = ω(xy) = ω((xy)∗) = ω(y∗x∗) = ω(y∗)+ω(x∗) = ω(y)+ω(x).
46 On division rings with general involution
Let R = {x ∈ D|ω(x) ≥ 0}. Evidently R is a subring of D, R is
total (i.e., contains x or x−1 for every non-zero element x in D), and
R is symmetric (i.e., contains εx∗x−1 for every element x in D ). For,
ω(εx∗x−1) = ω(εx∗) + ω(x−1) = ω(x) + ω(x−1) = 0.
Definition 3.1. Any subring R of D which is total and symmetric is
called ε-valuation ring.
Theorem 3.2. Given any ε-valuation ring R of a division ring D with
ε-involution, there exist a linearly ordered abelian group G , and an ε-
valuation ω : D → G such that R coincides with the valuation ring of ω.
Proof. Let U denote the multiplicative group of invertible elements of R.
Then εx∗x−1 ∈ U for every x ∈ D, because (εx∗x−1)−1 = x(x∗)−1ε−1 =
ε(εx∗)∗(εx∗)−1 ∈ R. Observe that ε and ε∗ are in U . Also, since U is
multiplicative, it follows that x∗x−1 = ε∗εx∗x−1 ∈ U , for every x ∈ D .
We now claim that xyx−1y−1 ∈ U for every x, y ∈ D. For, one has
the following identity
xyx−1y−1 = [ε(x∗)∗(x∗)−1][(εy∗x)∗(εy∗x)−1][εy∗y−1].
Hence, U contains the commutator subgroup [D, D] of D•. Thus U is
a normal subgroup of D•, and consequently, the factor group G = D•/U
is abelian. Define ω : D• → G to be the canonical mapping, and G order
by setting
a ∈ D•, aU ≥ 1 if and only if ω(a) ≥ 0.
Switching from the multiplicative linearly ordered group G to the
same ordered system where the multiplication is written addition, one
can quickly check axioms 1 to 4, so that, ω is an ε-valuation of D with
value group precisely G and with valuation ring evidently R.
Lemma 3.3. Any symmetric subring R of a division ring D with ε-
involution has the following properties:
1. R is closed under the ε-involution ( x ∈ R ⇒ x∗ ∈ R).
2. R contains ε, ε∗, and x∗x−1, for every x ∈ D .
3. R contains the commutator subgroup [D, D] , and hence R is pre-
served under conjugation (x−1Rx = R , for every x ∈ D );
4. Each ideal I of R is closed under the ε-involution, is two sided, and
is such that xy ∈ I implies yx ∈ I.
I. M. Idris 47
Proof. 1. If x ∈ R, then x∗ = ε∗.εx∗x−1.x ∈ R. Hence, R∗ = R. The
statements 2. and 3. follow from the proof of Theorem 3.2.
4. If I is a left ideal of R, and if x ∈ I, then x∗ = x∗x−1.x ∈ I ,
where x∗x−1 ∈ R. Also if y ∈ R, then x∗ ∈ I and εy∗ ∈ R together
imply that εy∗x∗ ∈ I, that is, (xyε∗)∗ ∈ I. Hence, ε−1xy = (xyε∗)∗∗ ∈ I.
However ε ∈ R and I is a left ideal of R, so that xy ∈ R. Thus I is also a
right ideal of R. Finally, if xy ∈ I, then x∗y∗ = (x∗x−1)(xy)(y−1y∗) ∈ I,
where x∗x−1, y−1y∗ ∈ R . Thus, yx = ε(x∗y∗)∗ε−1 ∈ I.
The following lemma is evident.
Lemma 3.4. The set J of non-invertible elements in R is a proper ideal
of R that contains every other ideal. One has
J = {x ∈ R|ω(x) > 0}.
Then one can form the residue division ring D = R/J . The map ∗ of D
defined by (x + J)∗ = x∗ + J, x ∈ R is an ε-involution of D.
In [1], a necessary and sufficient condition is given for extending an
abelian valuation from a division ring D to the over division ring K. We
will solve here a ∗-version of this problem, where valuation is replaced
with ε-valuation. Along the lines of the solution to the problem given
in [1], we shall consider pairs (V, J), where V is a ∗-closed subring of
D such that V ⊃ [D, D], and J is a ∗-closed ideal of V . We note that,
since [D, D] is a group, every element of [D, D] is a unit in V . Therefore,
[D, D]∩J = φ. We shall write (V, J) ≤ (V ′, J ′) and say (V ′, J ′) dominates
(V, J) if V ⊆ V ′ and J ⊆ J ′. Clearly this is a partial ordering for the
collection of all such pairs (V, J).
Lemma 3.5. Let D be a division ring with ε-involution ∗, R is a ∗-closed
subring containing [D, D] and M a ∗-closed proper ideal of R. Then there
exist a *-closed subring V containing [D, D], and a ∗-closed ideal J such
that (V, J) is maximal among pairs dominating (R, M). Further, any
such maximal pair (V, J) is made up of the ε-valuation ring V in D with
maximal 2-sided ideal precisely J .
Proof. Clearly the pairs (V, J) dominating (R, M) form an inductive sys-
tem and by Zorn’s Lemma, one can find a maximal member (V, J). Since
V is closed under the ε-involution and preserved under conjugation, it
follows that εx∗x−1 ∈ V for all x ∈ D. From the maximality follows that
J is a maximal ideal of V , and by the localization at J , we can enlarge
V to a local ring. Then by maximality, V is in fact a local ring with
maximal ideal J . To complete the proof we must show that V is a total
subring and this can be done as in the case of a commutative valuation
subring, see [4].
48 On division rings with general involution
Theorem 3.6. Let D ⊂ K be any division ring extension with ε-involution.
Given any ε-valuation ω on D with ε-valuation subring V and maximal
ideal J , there is an extension ε-valuation ω′ of ω to K if and only if
J [K, K] is a proper ideal of V [K, K], that is, if and only if there is no
equation of the form
∑
i
aici = 1, ai ∈ J, ci ∈ [K, K]. (1)
Proof. If there is an extension ω′ of ω, and (1) holds, then 0 = ω′(1) ≥
min{ω′(ai), ω
′(ci)}. Since ai ∈ J , ω(ai) > 0, and ω′(ci) = 0 (for, ci
is a product of commutators), it follows that the right hand side of the
preceding inequality is strictly positive, which is a contradiction. Hence
no equation as in (1) holds true. Then J [K, K] is a proper ideal of
V [K, K].
To prove the converse, we first note that J [K, K] = [K, K]J and
V [K, K] = [K, K]V (for, r(xyx−1y−1) = (rxyrr−1x−1y−1r−1)r). Since
both V and [K, K] are closed under the ε-involution, it follows that
(V [K, K])∗ = [K, K]∗V ∗ = [K, K]V = V [K, K], that is, V [K, K] is ∗-
closed. Similarly, J [K, K] is closed under the ε-involution. Now, clearly
V [K, K] ⊃ [K, K] and, hence, by Lemma 3.5, there is a maximal pair
(V ′, J ′) dominating (V, J). Thus, V ′ is an ε-valuation ring of K, which
defines the desired extension (by Theorem 3.2).
References
[1] K. P. M. Cohn and M. Mahdavi-Hezavehi, Extensions of valuations on skew fields,
Lecture Notes in Mathematics, 825, Springer-verlag, Berlin, 1980, 28-41.
[2] I. N. Herstein, Topics in Ring Theory, University of Chicago Press, Chicago, 1967.
[3] W. Scharlau, Zur Existenz von Involutionen auf einfachen Algebren, Math. Z. 145
(1975), 25-32.
[4] O. Schilling, The Theory of Valuations, Math. Surveys 4, Amer. Math. Soc.,
Providence, RI, 1950.
Contact information
I. M. Idris Department of Mathematics, Faculty of Sci-
ence, Ain-Shams University, Cairo 11566,
Egypt
E-Mail: idris@asunet.shams.edu.eg
Received by the editors: 24.04.2007
and in final form 24.05.2007.
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