Derivations on Pseudoquotients
A space of pseudoquotients denoted by B(X, S) is defined as equivalence classes of pairs (x, f); where x is an element of a nonempty set X, f is an element of S; a commutative semigroup of injective maps from X to X; and (x, f) ~ (y, g) for gx = fy: If X is a ring and elements of S are ring homomorp...
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irk-123456789-1655782020-02-15T01:25:55Z Derivations on Pseudoquotients Majeed, A. Mikusiński, P. Короткі повідомлення A space of pseudoquotients denoted by B(X, S) is defined as equivalence classes of pairs (x, f); where x is an element of a nonempty set X, f is an element of S; a commutative semigroup of injective maps from X to X; and (x, f) ~ (y, g) for gx = fy: If X is a ring and elements of S are ring homomorphisms, then B(X, S) is a ring. We show that, under natural conditions, a derivation on X has a unique extension to a derivation on B(X, S): We also consider (α, β) -Jordan derivations, inner derivations, and generalized derivations. Введено означення простору псевдочасток B(X, S) як класів еквiвалентностi пар (x, f), де x — елемент непорожньої множини X, f — елемент комутативної напівгрупи S ін'єктивних відображень із X у X; та (x, f) ~ (y, g), якщо gx = fy. Якщо X — кільце та елементи S є гомоморфізмами кільця, то B(X, S) є кільцем. Показано, що за природних умов похідна на X має єдине розширення до похідної на B(X, S). Також розглянуто (α, β)-жорданові похідні, внутрішні похідні та узагальнені похідні. 2013 Article Derivations on Pseudoquotients / A. Majeed, P. Mikusiński // Український математичний журнал. — 2013. — Т. 65, № 6. — С. 863–869. — Бібліогр.: 4 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165578 512.5 en Український математичний журнал Інститут математики НАН України |
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Короткі повідомлення Короткі повідомлення Majeed, A. Mikusiński, P. Derivations on Pseudoquotients Український математичний журнал |
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A space of pseudoquotients denoted by B(X, S) is defined as equivalence classes of pairs (x, f); where x is an element of a nonempty set X, f is an element of S; a commutative semigroup of injective maps from X to X; and (x, f) ~ (y, g) for gx = fy: If X is a ring and elements of S are ring homomorphisms, then B(X, S) is a ring. We show that, under natural conditions, a derivation on X has a unique extension to a derivation on B(X, S): We also consider (α, β) -Jordan derivations, inner derivations, and generalized derivations. |
| format |
Article |
| author |
Majeed, A. Mikusiński, P. |
| author_facet |
Majeed, A. Mikusiński, P. |
| author_sort |
Majeed, A. |
| title |
Derivations on Pseudoquotients |
| title_short |
Derivations on Pseudoquotients |
| title_full |
Derivations on Pseudoquotients |
| title_fullStr |
Derivations on Pseudoquotients |
| title_full_unstemmed |
Derivations on Pseudoquotients |
| title_sort |
derivations on pseudoquotients |
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Інститут математики НАН України |
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2013 |
| topic_facet |
Короткі повідомлення |
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http://dspace.nbuv.gov.ua/handle/123456789/165578 |
| citation_txt |
Derivations on Pseudoquotients / A. Majeed, P. Mikusiński // Український математичний журнал. — 2013. — Т. 65, № 6. — С. 863–869. — Бібліогр.: 4 назв. — англ. |
| series |
Український математичний журнал |
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AT majeeda derivationsonpseudoquotients AT mikusinskip derivationsonpseudoquotients |
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2025-07-14T19:00:25Z |
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2025-07-14T19:00:25Z |
| _version_ |
1837650020385423360 |
| fulltext |
UDC 512.5
A. Majeed, P. Mikusiński (CIIT, Islamabad, Pakistan)
DERIVATIONS ON PSEUDOQUOTIENTS
ПОХIДНI НА ПСЕВДОЧАСТКАХ
A space of pseudoquotients, denoted by B(X,S), is defined as equivalence classes of pairs (x, f), where x is an element
of a nonempty set X, f is an element of S, a commutative semigroup of injective maps from X to X, and (x, f) ∼ (y, g)
if gx = fy. If X is a ring and elements of S are ring homomorphosms, then B(X,S) is a ring. We show that, under
natural conditions, a derivation on X has a unique extension to a derivation on B(X,S). We also consider (α, β)-Jordan
derivations, inner derivations, and generalized derivations.
Введено означення простору псевдочасток B(X,S) як класiв еквiвалентностi пар (x, f), де x — елемент непорожньої
множини X, f — елемент комутативної напiвгрупи S iн’єктивних вiдображень iз X у X та (x, f) ∼ (y, g), якщо
gx = fy. Якщо X — кiльце та елементи S є гомоморфiзмами кiльця, то B(X,S) є кiльцем. Показано, що за
природних умов похiдна на X має єдине розширення до похiдної на B(X,S). Також розглянуто (α, β)-жордановi
похiднi, внутрiшнi похiднi та узагальненi похiднi.
1. Introduction. Let X be a ring (or an algebra ) with the unit I. An additive (or linear) map δ
from X into it self is called a derivation if δ(AB) = δ(A)B +Aδ(B) for all A,B ∈ X. Derivations
are very important both in theory and applications, and are studied by many mathematicians. An
additive (or linear) map δ from a ring (or an algebra) X into itself is called a Jordan derivation if
δ(A2) = δ(A)A+Aδ(A) for all A ∈ X.
Let X be any nonempty set and S be a commutative semigroup acting on X injectively. This
means that every φ ∈ S is an injective map φ : X → X and (φψ)x = φ(ψx) for all φ, ψ ∈ S and
x ∈ S and x ∈ X. For (x, φ), (y, ψ) ∈ X x S we write (x, φ) (y, ψ) if ψx = φy.
It is easy to check that is an equivalence relation in X x S, finally we define
B(X,S) = (XxS)/∼. The equivalence class of (x, φ) will be denoted by
x
φ
. The set of psedo-
quotients.
This is a slight absue of notion, but we follow here the tradition of denoting rational numbers by
p
q
even through the same formal problem is present there.
Elements of X can be identified with elements of B(X,S) via the embedding ι : X → B(X,S)
defined by
ι(x) =
φx
φ
,
where φ is an arbitrary element of S, clearly is well defined that is, it is independent of φ. Action of
S can be extended to B(X,S) via
φ
x
ψ
=
φx
ψ
If φ
x
ψ
= i(y), for some y ∈ X, we will write φ
x
ψ
∈ X and φ
x
ψ
= y, which formally incorrect, but
convenient and harmless. For instance, we have φ
x
φ
= x.
c© A. MAJEED, P. MIKUSIŃSKI, 2013
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6 863
864 A. MAJEED, P. MIKUSIŃSKI
Element of S, when extended to maps on B(X,S), become bijections. The action of ψ−1 on
B(X,S) can be defined as
ψ−1x
φ
=
x
φψ
.
Consequently, S can be extended to a commutative group of bijections acting on B(X,S).
If (X,�) is a commutative group and S is a commutative semigroup of injective homomorphsims
on X, then B(X,S) is a commutative group with the operation defined as
x
φ
� y
ψ
=
ψx� φy
φψ
.
Similarly, if X is a vector space and S is a commutative semigroup of injective linear mapping
from X into X, then B(X,S) is a vector space with the operation defined as
x
φ
+
y
ψ
=
ψx+ φy
φψ
and λ
x
φ
=
λx
φ
.
If δ : X → X, if δ extends to a map δ̂ : B(X,S) → B(X,S), it is often important to know
what properties of δ are inherited by δ̂. In this section we consider some special situations when an
extension is possible, which are important for the particular case studied in this paper .
If δ(fx) = fδ(x) for all x ∈ X and all f ∈ S, then we say that δ commutes with S.
The following Proposition 1.1 in [1] is use full to prove the following theorems.
Proposition 1.1. Let δ : X → X. Then
δ̂
(
x
f
)
=
δ(x)
f
is a well-defined extension of δ to δ̂ : B(X,S)→ B(X,S) if and only if δ commutes with S.
2. Derivation on pseudoquotients. In this section we study about extension of (α, β)-derivations
on B(X,S). And show under certain conditions it commutes with f is an injective ring homomor-
phisms form set S on X. Where S is a commutative semigroup of injective ring homomorphisms.
2.1. (α, β)-Derivations. Let X be a ring and let α and β be endomorphisms of X. By an
(α, β)-derivation on X we mean a map δ : X → X such that
δ(xy) = δ(x)β(y) + α(x)δ(y) for all x, y ∈ X.
A (1, 1)-derivation, where 1 is the identity map on X is called simply a derivation. That is, by a
derivation we mean a map δ : X → X such that
δ(xy) = δ(x)y + xδ(y) for all x, y ∈ X.
Theorem 2.1. Let X be a ring and let S be a commutative semigroup of injective ring ho-
momorphisms. Let α and β be homomorphisms from X into itself that commute with S, that is,
αf(x) = fα(x) and βf(x) = fβ(x) for every f ∈ S and x ∈ X. If δ is an (α, β)-derivation on X
that commutes with S, then the map δ̂ : B → B defined by
δ̂
(
x
f
)
=
δ(x)
f
(2.1)
is an extension of δ to an (α, β)-derivation on B.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
DERIVATIONS ON PSEUDOQUOTIENTS 865
Proof. Assume δ is an (α, β)-derivation on X that commutes with S. Then δ̂ is well-defined by
Proposition 1.1 in [1]. In order to show that it is an (α, β)-derivation on B, consider
x
f
,
y
g
∈ B(X,G).
Then
δ̂
(
x
f
y
g
)
=
δ(gxfy)
fg
=
δ(gx)β(fy) + α(gx)δ(fy)
fg
=
=
δx
f
β(y)
g
+
α(x)
f
δy
g
= δ̂
(
x
f
)
β
(
y
g
)
+ α
(
x
f
)
δ̂
(
y
g
)
.
Theorem 2.1 is proved.
Corollary 2.1. Let X be a ring and let S be a commutative semigroup of injective ring ho-
momorphisms. If δ is a derivation on X that commutes with S, then the map δ̂ : B → B defined
by
δ̂
(
x
f
)
=
δ(x)
f
is an extension of δ to a derivation on B.
2.2. (α, β)-Jordan derivations. Let α and β be endomorphisms of X. By an (α, β)-Jordan
derivation on X we mean a map δ : X → X such that
δ(x2) = δ(x)β(x) + α(x)δ(x) for all x ∈ X.
A (1, 1)-Jordan derivation, where 1 is the identity map on X is called simply a Jordan derivation.
That is, by a Jordan derivation on X we mean a map δ : X → X such that
δ(x2) = δ(x)x+ xδ(x) for all x ∈ X.
Theorem 2.2. Let X be a ring and let S be a commutative semigroup of injective ring ho-
momorphisms. Let α and β be homomorphisms from X into itself that commute with S, that is,
αf(x) = fα(x) and βf(x) = fβ(x) for every f ∈ S and x ∈ X. If δ is an (α, β)-Jordan derivation
on X that commutes with S, then the map δ̂ : B → B defined by
δ̂
(
x
f
)
=
δ(x)
f
is an extension of δ to an (α, β)-Jordan derivation on B.
Proof. The proof is similar to the proof of Theorem 2.1.
Corollary 2.2. Let X be a ring and let S be a commutative semigroup of injective ring ho-
momorphisms. If δ is a derivation on X that commutes with S, then the map δ̂ : B → B defined
by
δ̂
(
x
f
)
=
δ(x)
f
is an extension of δ to a derivation on B.
In Theorem 2.2 and the above corollary it is necessary to assume that δ commutes with S. The
next theorem describes a situation which guarantees that δ commutes with S.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
866 A. MAJEED, P. MIKUSIŃSKI
Theorem 2.3. Let X be an unital Banach algebra and let f be an injective algebra homo-
morphism. Let α and β be algebra homomorphisms from X into itself that commute with f, if δ is a
linear mapping on X such that
δ(xx−1) = α(x)δ(x−1) + δ(x)β(x−1) (2.2)
for every invertible element x ∈ X, then δ is an (α, β)-Jordan derivation on X and commutes with f.
Proof. It is known that (2.2) implies δ(e) = 0, where e is the identity element in X. Therefore,
δ(fe) = 0 for any f injective homomorphism onX. In order to show that linear mapping δ is a Jordan
derivation and commutes with S we have to show that δfy2 = fδy2. For any T in X. Let n be a
positive integer with n > ‖T‖+e and y = ne+T. We have that y and e−y are invertible in X. Since
α(fx−1) = α(fx)−1 = fα(x−1) = fα(x−1) and β(fx−1) = β(fx)−1 = fβ(x−1) = fβ(x−1) for
any invertible element x in X. Then
δ(fy) = −α(fy)δ(fy−1)β(fy) = −α(fy)δ(fy−1f(e− y)2 − fy)β(fy) =
= α(fy)α(fy−1f(e− y)2)δ(f(e− y)−2fy)β(y−1(e− y)2)β(fy) + α(fy)δ(fy)β(fy) =
= α(fy)α(fy−1 − 2fe+ fy)δ(f(e− y)−2 − f(e− y)−1)β(fy−1 − 2fe+ fy)β(fy)+
+α(fy)δ(fy)β(fy) =
= (e− 2α(fy) + αf(y)2)δ(f(e− y)−2 − f(e− y)−1)(e− 2β(fy) + βf(y2))+
+α(fy)δ(fy)β(fy) = α(f(e− y)2)δ(f(e− y)−2)β(f(e− y)2)−
−(αf(e− y))2)δ((e− y)−1)(βf(e− y))2 + α(fy)δ(fy)β(fy) =
= −δ(f(e− y)2) + αf(e− y)δf(e− y)βf(e− y) + α(fy)δ(fy)β(fy) =
= 2δ(fy)− δf(y2)− δ(fy) + α(fy)δ(fy) + δ(fy)β(fy)−
−α(fy)δ(fy)β(fy) + α(fy)δ(fy)β(fy) =
= δ(fy)− δ(fy2) + α(fy)δ(fy) + δ(fy)β(fy).
Hence δ(fy2) = δ(fy)β(fy) + α(fy)δ(fy). Since δ(fe) = 0 and fy = f(ne) + ft, we have that
δ(ft2) = δ(ft)β(ft) + α(ft)δ(ft) for any t ∈ X. Similarly we can show for fδx.
Theorem 2.3 is proved.
Corollary 2.3. Let X be an unital Banach algebra and let f be an injective algebra homomor-
phism on X. Let α and β be algebra homomorphisms from X onto itself that commute with f. If δ
is a linear mapping on X such that
δ(xx−1) = α(x)δ(x−1) + δ(x)β(x−1)
for every invertible element x ∈ X, then δ is an (α, β)-Jordan derivation on X that commutes with
S = {fn : n = 1, 2, 3, . . .} and the map δ̂ : B(X,S)→ B(X,S), defined by
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
DERIVATIONS ON PSEUDOQUOTIENTS 867
δ̂
(
x
f
)
=
δ(x)
f
,
is an extension of δ to an (α, β)-Jordan derivation on B(X,S).
2.3. Idempotent. An idempotent element of a ring is an element which is idempotent with respect
to the ring’s multiplication, that is, r2 = r. A ring in which all elements are idempotent is called a
Boolean ring.
Lemma 2.1. Let X be a ring and let S be a commutative semigroup of injective ring homo-
morphisms.
x
f
is idempotent in B(X,S) if and only if x is idempotent in X.
Proof. If
x
f
is idempotent in B(X,S), then
x
f
=
x
f
x
f
=
fxfx
f2
=
f(x2)
f2
=
x2
f
.
Consequently, x = x2. The proof in the other direction follows from the above.
Theorem 2.4. Let X be a commutative ring and let S be a commutative semigroup of injective
ring homomorphisms and let δ be a derivation on X that commutes with S. If δ̂ is the extension of δ
onto B(X,S) as defined by (2.1) and
x
f
∈ B(X,S) is idempotent, then
(i) δ̂
(
x
f
)
= 0,
(ii) δ̂
(
y
g
x
f
)
= δ̂
(
y
g
)
x
f
for any
y
g
∈ B(X,S),
(iii) δ̂
(
x
f
y
g
)
=
x
f
δ̂
(
y
g
)
for any
y
g
∈ B(X,S).
Proof. For any idempotent
x
f
∈ B(X,S), we have
δ̂
(
x
f
)
= δ̂
(
x
f
x
f
)
= δ̂
(
x
f
)
x
f
+
x
f
δ̂
(
x
f
)
.
As X is a commutative ring,
= δ̂
(
x
f
)
x
f
+ δ̂
(
x
f
)
x
f
and consequently
δ̂
(
x
f
)
x
f
= δ̂
(
x
f
)
x
f
+ δ̂
(
x
f
)
x
f
.
This shows that δ̂
(
x
f
)
= 0.
(ii) δ̂
(
y
g
x
f
)
= δ̂
(
y
g
)
x
f
+
g
y
δ̂
(
x
f
)
=
g
y
δ̂
(
x
f
)
.
Similarly we can show δ̂
(
x
f
y
g
)
=
x
f
δ̂
(
y
g
)
.
Theorem 2.4 is proved.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
868 A. MAJEED, P. MIKUSIŃSKI
By induction, it is easy to show that for any idempotents
x1
f
,
x2
f
, . . . ,
xn
f
∈ B and any
y
g
∈ B,
δ̂
(
x1
f
x2
f
. . .
xn
f
y
g
)
=
x1
f
x2
f
. . .
xn
f
δ̂
(
y
g
)
.
2.4. Inner derivations. An inner derivation on X is a map δ : X → X such that
δ(x) = xy − yx for each y ∈ X.
Let X be a ring and let S be a commutative semigroup of injective ring homomorphisms.
δ : X → X is an inner derivation for each x ∈ X and for each f ∈ S
δ(x) = xf − fx.
Theorem 2.5. Let X be a ring and let S be a commutative semigroup of injective ring ho-
momorphisms. If δ is a inner derivation on X, then the map δ̂ : B → B defined by δ̂
(
x
f
)
=
=
2δ(x)
f
− δ(fx)
f2
is an extension of δ to a inner derivation on B if xf − fx commutes with S for
every f ∈ S.
2.5. Generalized derivation. δ : X → X is a map on X is called a generalized derivation if there
exists a derivation d : X → X such that
δ(xy) = δ(x)y + xd(y) for all x, y ∈ X.
Theorem 2.6. Let X be a ring and let S be a commutative semigroup of injective ring homo-
morphisms. If δ is a generalized derivation on X, then the map δ̂ : B → B defined by
δ̂
(
x
f
)
=
δ(x)
f
is an extension of δ to a generalized derivation on B.
Proof. Assume that δ and d commutes with S. In order to show that it is an δ is a generalized
derivation on B, consider
x
f
,
y
g
∈ B(X,G):
δ̂
(
x
f
y
g
)
=
δ(gxfy)
fg
=
δ(gx)(fy) + (gx)d(fy)
fg
=
=
δx
f
y
g
+
x
f
dy
g
= δ̂
(
x
f
)(
y
g
)
+ α
(
x
f
)
d̂
(
y
g
)
.
Theorem 2.6 is proved.
Example 2.1. Let R be a commutative ring and let δ be a derivation on R. For an element
x ∈ R we denote by Mx the homomorphism defined by Mx(y) = xy. Let
S = {Mx : x ∈ R,Mx is injective, and δ(x) = 0} .
Since
δ(Mx(y)) = δ(xy) = δ(x)y + xδ(y) = xδ(y) =Mx(δy)
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
DERIVATIONS ON PSEUDOQUOTIENTS 869
for every Mx ∈ S, δ can has a unique extension to a derivation on B(R,S).
For a simple example we can take for R the ring of polynomials in x and y and δ =
∂
∂y
. Then
S is not trivial and, since it contains homomorphism that are not surjective, B(R,S) is a nontrivial
extension of R.
Example 2.2. Let N be a nest algebra and S be a commutative semigroup acting on N gener-
ated by finite rank operators. δ is a derivation on N with δ(φ) = 0.
Let for any arbitrary n from N and φ from G. From [4] every finite rank operator in N rep-
resented as a sum of rank one operators. From [3] Every rank one operator in N denoted as linear
combination of at most four idempotents.
Hence we have δ(φn) = φδ(n) + nδ(φ). Where δ(φ) = 0, for every rank one operator from S.
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Received 23.04.12
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
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