EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 13, No. 3, 2020, 472-482
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Adjunction and Localization in the Category A-Alg of
A-Algebras

Moussa Thiaw1,∗, Mohamed Ben Faraj Ben Maaouia1

1 UFR-SAT/Gaston Berger, University, Saint-Louis, Senegal

In memory of the dead of COVID-19

Abstract. In our paper [3] we built the functor Êxt
n

S−1A(-, S−1B) in the category A-Alg. The
purpuse of this paper is to show that if A is a ring not necessary commutative, S a central
multiplicatively closed subset of A and B an (A-A)-bialgebra, then

TorS
−1A

n (-, S−1B) : Alg-S−1A� S−1A-Modo : Êxt
n

S−1A(-, S−1B)o

is an adjunction.

2020 Mathematics Subject Classifications: 18A25, 18A40, 18N40, 16E35

Key Words and Phrases: Adjunction, Localization, Algebra, Functor

1. Introduction

In this paper, A is assumed unitary, associative and not necessarily commutative. A
and B are algebras assumed unitary, associative and not necessarily commutative as a
ring and unital as an A-module. In general, the action of the functor Êxt

n

A(-,B) on an
A-module M(resp. A-algebra A ) is not an algebra. In our paper [3] we built the functors

Êxt
n

A(-,B) : Alg-A→ B-Alg and S−1() : A-Alg → S−1(A)-Alg. The notion of adjunction
allows to see if two categories are equivalent. This notion of adjunction makes it possible
to preserve some properties from one category to another, such as monomorphisms. The
purpose of this paper is to show that TorS

−1A
n (-, S−1B) : Alg-S−1A → S−1A-Modo and

Êxt
n

S−1A(-, S−1B)o : S−1A-Modo → Alg-S−1A are adjoint functors, where S is a central
multiplicatively closed subset of A and B a (A-A)-bialgebra but before, we show that the
functors - ⊗A S−1(A) : Alg-A → A-Modo and HomA(-, S−1(A)) : A-Modo → Alg-A are
adjoint, then we show that the functors S−1() : A-Alg → S−1(A)-Alg and - ⊗A S−1(A) :

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v13i3.3742

Email addresses: thiawskr@gmail.com (M. Thiaw),
mohamed-ben.maaouia@ugb.edu.sn, maaouiaalg@hotmail.com (M. Maaouia)

https://www.ejpam.com 472 c© 2020 EJPAM All rights reserved.



M Thiaw, M Maaouia / Eur. J. Pure Appl. Math, 13 (3) (2020), 472-482 473

Alg-A→ A-Modo are naturally isomorphic and we deduce that S−1() : Alg-A� A-Modo :
HomA(-, S−1(A))o is an adjunction. This paper is divided into three parts. In the first part
entitled ”preliminary results” we recall some basic results. In the second part, we show
that the functors S−1() : Alg-A → A-Modo and HomA(-, S−1(A))o : A-Modo → Alg-A
are adjoint and in the third part we show the main results of this paper (see Theorem 6).

2. Preliminary Results

Proposition 1. Let A and A be two rings, and θ : A −→ A be a ring morphism. Then
A has a structure of left (resp. right) A-module as the same way:

• : A×A −→ A
(a, x) 7−→ a • x = θ(a)x

(resp. ∗ : A ×A −→ A
(x, a) 7−→ x ∗ a = xθ(a))

Proof. Easy.

In all that follows • (resp. ∗) designates the external law of the left (resp. right)
A-module A relatively to θ.

Definition 1. Let A and A be two rings, and θ : A −→ A be a ring morphism. Then
(A ,+,×, •) (resp. (A ,+,×, ∗)) is called left (resp. right) A-algebra relatively to θ.

This definition shows that to provide A with a structure of left (resp. right) A-algebra
it suffices to have a ring morphism of A into A .

Definition 2. [1] Let A and A be two rings, and θ : A −→ A be a ring morphism. If
Im(θ) ⊆ Z(A ), then A is called an A-algebra relatively to θ.

Definition 3. Let A be a ring, a subset S of A is called multiplicative if 1A ∈ S and S is
stable by multiplication i.e for all x, t ∈ S, xt ∈ S.

Definition 4. Let A be a ring and S a multiplicative subset of A. We say that S is closed
if for all s, s′ ∈ A such that ss′ ∈ S ⇒ s ∈ S and s′ ∈ S.

Definition 5. Let S be a multiplicatively closed subset of a ring A. We say that S satisfies
the left Ore conditions if:

(i) ∀a ∈ A, ∀s ∈ S ∃t ∈ S and b ∈ A such that ta = bs

(ii) ∀a ∈ A, ∀ s ∈ S such that as = 0, then it exist t ∈ S such that ta = 0.

Theorem 1. Let A be a ring and S a multiplicatively closed subset of A satisfying the left
Ore conditions. The binary relation defined in S ×M by

(s,m)R(s′,m′)⇐⇒ ∃x, y ∈ S :

{
xm = ym′

xs = ys′

is an equivalence relation.



M Thiaw, M Maaouia / Eur. J. Pure Appl. Math, 13 (3) (2020), 472-482 474

Proof. See [4], [2] and [5].

Theorem 2. Let A be a ring not necessary commutative and S a multiplicatively closed
subset of A satisfying the left Ore condition, then S−1A is a ring by the two following
operations:

• a
t + b

s = xa+yb
ys where x, y ∈ S : xt = ys

• a
t ×

b
s = zb

wt where (w, z) ∈ S ×A : wa = zs.

Proof. see [4].

Theorem 3. Let A be a left (resp. right) A-algebra and S a central multiplicatively closed
subset of A. Then S−1(A ) ∈ Ob(S−1A-Alg) (resp. S−1A ∈ Ob(Alg-S−1A)).

Proof. See [3].

Definition 6. Let F,G : C → D be two covariant functors. A natural transforma-
tion θ from F to G is an assignment to every object X of C of a morphism θX ∈
HomD(F (X), G(X)) such that for any morphism f ∈ HomC (X,Y ), the following dia-
gram commutes in D

F (X)

F (f)
��

θX // G(X)

G(f) ⇔ G(f)◦θX=θY ◦F (f).
��

F (Y )
θY
// G(Y )

If θX is an isomorphism for any object X of C , then θ is called functorial isomorphism .

Notation: We note by F ∼= G, if there is a functorial isomorphism θ : F → G.

Definition 7. A pair of functors F : C � D : G is an adjunction if we have for any
(X,Y ) ∈ Ob(C )×Ob(D) a bijection

φX,Y : HomD(FX, Y ) −→ HomC (X,GY )

such that the following two squares are commutative:

HomD(FX ′, Y )
φX′,Y //

(Ff)∗

��

HomC (X ′, GY )

f∗ and

��
HomD(FX, Y )

φX,Y // HomC (X,GY )

HomD(FX, Y )
φX,Y //

g∗
��

HomC (X,GY )

(Gg)∗
��

HomD(FX, Y ′)
φX,Y ′ // HomC (X,GY ′)

where f : X −→ X ′ and g : Y −→ Y ′ are morphisms.



M Thiaw, M Maaouia / Eur. J. Pure Appl. Math, 13 (3) (2020), 472-482 475

Proposition 2. If F : C −→ D has two right (resp. left) adjoint G and H, then G and
H are naturally isomorphic.
Reciprocally, if F is left (resp. right) adjoint to G, and G is naturally isomorphic to H,
then F is also a left adjoint to H.

Proposition 3. Suppose Ki is the i-th syzygy of some projective resolution of B. Then

Torn+1(-,B) and Torn−i(-,Ki) are naturally isomorphic functors; also Êxt
n+1

(-,B) and

Êxt
n−i

(-,Ki) are naturally isomorphic functors.

Proof. See [6, chap. 10] and [7, chap. 5].

3. The Adjunction of the Functors S−1() and HomA(-,B) in the Category
of A-Alg

Theorem 4. Let A be a ring and S a central multiplicatively closed subset of A. Then
the functors S−1() and -⊗A S−1(A) are naturally isomorphic (S−1() ∼= -⊗A S−1(A) ).

Proof. Let θ : S−1()→ -⊗A S−1(A).
∗ Show that θ is a natural transformation.
Let A ∈ Ob(A-Alg).
Consider

θA : A × S−1(A) −→ S−1(A )

(x,
a

s
) 7−→ a · x

s

We have θA which is A-bilinear, so by the universal property of tensor product, there exist
a morphism of groups θA : A ⊗ S−1(A) −→ S−1(A ) defined by

θA (xi ⊗
∑ ai

si
) =

∑ ai · xi
si

.

Let f ∈ HomA-Alg(A ,A ′), show that the following diagram is commutative

A ⊗ S−1(A)

f⊗1S−1(A)

��

θA // S−1(A )

S−1(f) ⇔ S−1(f)◦θA =θA ′◦(f⊗1S−1(A)).

��
A ′ ⊗ S−1(A)

θA ′

// S−1(A ′)

Let xi ⊗
∑ ai

si
∈ A ⊗ S−1(A).

On the one hand we have,

θA ′ ◦ f ⊗ 1S−1(A)(xi ⊗
∑ ai

si
) = θA ′(f(xi)⊗

∑ ai
si

) =
∑ ai · f(xi)

si
.

On the other hand



M Thiaw, M Maaouia / Eur. J. Pure Appl. Math, 13 (3) (2020), 472-482 476

S−1(f) ◦ θA (xi ⊗
∑ ai

si
) = S−1(f)(

∑ ai · xi
si

) =
∑
S−1(f)(

ai · xi
si

) =
∑ ai · f(xi)

si
.

So we have, S−1(f) ◦ θA = θA ′ ◦ (f ⊗ 1S−1(A)) and therefore θ is a natural transformation.

∗ Show that for any A ∈ Ob(A-Alg), θA is bijective.

• Let
x

s
∈ S−1(A ).

We have θA (x⊗ 1

s
) =

x

s
⇒ θA is surjective.

• Let xi ⊗
∑ ai

si
∈ A ⊗ S−1(A).

We have ∑
xi ⊗

ai
si

=
∑

(
∏

si)siaixi ⊗
1∏
si
.

Pose

s =
∏
si and zi = s−1si ∈ S.

So we have ∑
xi ⊗

ai
si

=
∑

ziaixi ⊗
1

s
= (
∑

ziaixi)⊗
1

s
.

So the elements of S−1(A)⊗A are written in the form
1

s
⊗ Y , where Y ∈ A and s ∈ S.

Let
1

s
⊗ Y ∈ KerθA ⇔ θA (

1

s
⊗ Y ) = 0S−1A ⇒

Y

s
=

0A

1
⇒ ∃ s1, s2 ∈ S such that{

s1Y = 0
s1s = s2

So
1

s
⊗ Y =

1

ss1
⊗ s1Y =

1

ss1
⊗ 0 = 0 ⇒ KerθA = {0S−1A } ⇒ θA is injective.

So θA is bijective.

Therefore S−1() ∼= -⊗A S−1(A).

Theorem 5. Let A be a ring and S a central multiplicatively closed subset of A. Then

-⊗A S−1(A) : Alg-A� A-Modo : HomA(-, S−1(A))o

is an adjunction.

Proof. ∗ Let A ∈ Ob(Alg-A), R ∈ Ob(A-Modo).

Let ϕA ,R : HomA-Modo(A ⊗A S−1(A),R) −→ HomAlg-A(A , HomA(R, S−1(A))o)
f 7−→ ϕA ,R(f) : A −→ HomA(S−1(A),R)

x 7−→ ϕA ,R(f)(x) : S−1(A) −→ R
y

s
7−→ f(x⊗ y

s
).



M Thiaw, M Maaouia / Eur. J. Pure Appl. Math, 13 (3) (2020), 472-482 477

It is clear that ϕA ,R is well defined.
Consider the map,

ψ : HomAlg-A(A , HomA(R, S−1(A))o) −→ HomA-Modo(A ⊗A S−1(A),R)
g 7−→ ψ(g) : A ⊗A S−1(A) −→ R

x⊗ y

s
7−→ ψ(g)(x⊗ y

s
) = g(x)(

y

s
).

Let g ∈ HomAlg-A(A , HomA(R, S−1(A))o), x ∈ A ,
y

s
∈ S−1(A),

we have ϕA ,R ◦ ψ(g)(x⊗ y

s
) = ϕA ,R(ψ(g)(x⊗ y

s
)) = ϕA ,R(g(x)(

y

s
)) = g(x⊗ y

s
).

Hence

ϕA ,R ◦ ψ(g) = g, ∀ g ∈ HomAlg-A(A , HomA(R, S−1(A))o).

So
ϕA ,R ◦ ψ = 1HomAlg-A(A ,HomA(R,S−1(A))o).

Similarly, we show that ψ ◦ ϕA ,R = 1HomAlg-A(A⊗S−1(A),R).
So

ϕA ,R is an isomorphism of left A-algebras.

∗ It remains to show that ϕA ,R is natural in A and in R.
Let f : A −→ A ′ and g : R −→ R′ be two morphisms of left A-algebras.
Pose

F = -⊗A S−1(A) and G = HomA(-, S−1(A))o.

We have f∗ ◦ ϕA ′,R(h)(x)(
y

s
) = ϕA ′,R(h) ◦ f(x)(

y

s
) = ϕA ′,R(h)(f(x))(

y

s
)

= h(f(x)⊗ y

s
).

On the other hand we have:
ϕA ,R ◦ (Ff)∗(h)(x)(

y

s
) = (Ff)∗(h)(x⊗ y

s
) = h(Ff)(x⊗ y

s
) = h(f ⊗ 1S−1(A))(x⊗

y

s
)

= h(f(x)⊗ y

s
).

So

f∗ ◦ ϕA ′,R(h)(x)(
y

s
) = ϕA ,R ◦ (Ff)∗(h)(x)(

y

s
), ∀ h ∈ HomA(A ⊗ S−1(A),R), x ∈ A

and
y

s
∈ S−1(A).

So f∗ ◦ ϕA ′,R = ϕA ,R ◦ (Ff)∗ ⇒ ϕA ,R is natural in A .
We show in the same way that ϕA ,R is natural in R.
Therefore the functors -⊗A S−1(A) and HomA(-, S−1(A))o are adjoint.



M Thiaw, M Maaouia / Eur. J. Pure Appl. Math, 13 (3) (2020), 472-482 478

Corollary 1. Let A be a ring and S a central multiplicatively closed subset of A. Then

S−1() : Alg-A� A-Modo : HomA(-, S−1(A))o

is an adjunction.

Proof. By the theorem 4, we have -⊗A S−1(A) is a left adjoint to HomA(-, S−1(A))o

and by the theorem 4 S−1() is isomorphic to HomA(-, S−1(A))o, so the fonctor S−1() is a
left adjoint to HomA(-, S−1(A))o.

Corollary 2. Let A be a duo ring, P a prime ideal of A and S = (A-P ) ∩ Z(A). Then

S−1() : Alg-A� A-Modo : HomA(-, S−1(A))o

is an adjunction.

Proof. Since A is a duo ring, then A-P is a multiplicatively closed subset of A, so
S = (A-P ) ∩ Z(A) is a central multiplicatively closed subset of A.
So by the corollary 1 S−1() is a left adjoint to HomA(-, S−1(A))o.

Corollary 3. Let A be a duo ring, P a prime ideal of A, SR the set of regular elements
of A-P and S = SR ∩ Z(A). Then

S−1() : Alg-A� A-Modo : HomA(-, S−1(A))o

is an adjunction.

Proof. Since A is a duo ring, then the set of regular elements, SR, of A-P is a multi-
plicatively closed subset of A, so S = SR ∩Z(A) is a central multiplicatively closed subset
of A.
So by the corollary 1 S−1() is a left adjoint to HomA(-, S−1(A))o.

Proposition 4. Let A be a ring, B a (A-A)-bialgebra and S a central multiplicatively
closed subset of A. Then

-⊗A S−1(B) : Alg-S−1A� S−1A-Modo : HomA(-, S−1(B))o

is an adjunction.

Proof. The proof is similar to that of the previous theorem.



M Thiaw, M Maaouia / Eur. J. Pure Appl. Math, 13 (3) (2020), 472-482 479

4. The Adjunction of the Functors Êxt
n

S−1A(-, S
−1B) and TorS

−1A
n (-, S−1B)

in the Category A-Alg

Proposition 5. Let B be a (B-A)-bialgebra.Then the correspondence

Êxt
n

B(-,B) : B-Mod −→ Alg-A

(i) which has any left B-module M , we associate the right A-algebra Êxt
n

B(M,B),

(ii) which has any morphism of left B-modules f : M −→M ′, we associate Êxt
n

B(f,B) :

Êxt
n

B(M ′,B)→ Êxt
n

B(M,B)

is a contravariant functor.

Proof. ∗ We have M ∈ Ob(B-Mod) ⇒ Êxt
n

B(M,B) ∈ Ob(Alg-A) (see [3]), so the

action of Êxt
n

B(-,B) on the objects of Alg-A makes sense.

∗ Let f : M → M ′ be a morphism of left B-modules. By the comparison theorem
we have the following commutative diagram

PM : . . . //

f̃
��

Pn //

f̃n
��

Pn−1 · · · //

˜fn−1

��

P0
//

f̃0
��

A //

f

��

0

PM ′ : . . . // P ′n // P ′n−1 · · · // P ′0
// B // 0

By applying the contravariant functor HomB(-,B) we have

HomB(PM ,B) : 0 //

HomB(f̃ ,B)
��

HomB(M,B) //

HomB(f,B)
��

HomB(P0,B) · · ·

HomB(f̃0,B)
��

HomB(PM ′ ,B) : 0 // HomB(M ′,B) // HomB(P ′0,B) · · ·

So HomB(f̃ ,B) : HomB(PM ,B) −→ HomB(PM ′ ,B) is a morphism of chain complex.
We have

Hn(HomB(f̃ ,B)) : Hn(HomB(PM ,B)) −→ Hn(HomB(PM ′ ,B))

zn 7−→ HomB(f̃n,B)zn.

HomB(f,B) = f∗ and HomB(f̃n,B) = f̃∗n are morphisms of right A-algebras (see [3]).

So Hn(HomB(f̃ ,B)) = Êxt
n

B(f,B) is a morphism of right A-algebras, so the action of

Êxt
n

B(-,B) on the arrow makes sense.
∗ We have

Êxt
n

B(g ◦ f,B) = Hn(HomB( ˜g ◦ f,B)) = Hn(HomB(g̃ ◦ f̃ ,B))
= Hn[HomB(f̃ ,B) ◦HomB(g̃,B)]
= Hn(HomB(f̃ ,B)) ◦Hn(HomB(g̃,B))

= Êxt
n

B(f,B) ◦ Êxt
n

B(g,B).



M Thiaw, M Maaouia / Eur. J. Pure Appl. Math, 13 (3) (2020), 472-482 480

∗ We have Êxt
n

B(1M ,B)(zn) = HomB((1̃M )n,B)(zn) = 1HomB(M,B)(zn) = zn.
So

Êxt
n

B(1M ,B) = 1
Êxt

n

B(M,B)
.

Therefore Êxt
n

B(-,B) : B-Mod −→ Alg-A is a contravariant functor.

Proposition 6. Let B be a (B-A)-bialgebra.Then the correspondence

Êxt
n

B(-,B)o : B-Modo −→ Alg-A

(i) which has any left B-module M , we associate the right A-algebra Êxt
n

B(-,B)o(M) =

Êxt
n

B(M,B),

(ii) which has any f ∈ HomB-Modo(M,M ′), we associate Êxt
n

B(-,B)o(f) = Êxt
n

B(f,B).

is a covariant functor.

Proof. ∗ LetM ∈ Ob(B-Modo), we have Êxt
n

B(-,B)o(M) = Êxt
n

B(M,B) ∈ Ob(Alg-A),

so the action of Êxt
n

B(-,B)o on the objects of B-Modo makes sense.
∗ Let f ∈ HomB-Modo(M,M ′).

SinceHomB-Modo(M,M ′) = HomB-Mod(M
′,M), then f ∈ HomB-Mod(M

′,M), so Êxt
n

B(f,B) ∈
HomAlg-A(Êxt

n

B(M,B), Êxt
n

B(M ′,B)) because Êxt
n

B(f,B) is a contravariant functor.
And hence

Êxt
n

B(-,B)o(f) = Êxt
n

B(f,B) ∈ HomAlg-A(Êxt
n

B(M,B), Êxt
n

B(M ′,B)).

Therefore the action of Êxt
n

B(-,B)o on the arrow makes sens.
∗ Let f ∈ HomB-Modo(M,M ′), g ∈ HomB-Modo(M ′,M ′′).
We have

Êxt
n

B(g ◦ f,B)o = Êxt
n

B(g ◦ f,B) = Êxt
n

B(f,B) ◦ Êxt
n

B(g,B)

= Êxt
n

B(f,B)o ◦B-Modo Êxt
n

B(g,B)o

= Êxt
n

B(g,B)o ◦B-Mod Êxt
n

B(f,B)o.

Therefore Êxt
n

B(-,B)o : B-Modo −→ Alg-A is a covariant functor.

Theorem 6. Let A be a ring, S a central multiplicatively closed subset of A and B a
(A-A)-bialgebra. Then

TorS
−1A

n (-, S−1B) : Alg-S−1A� S−1A-Modo : Êxt
n

S−1A(-, S−1B)o

is an adjunction.



M Thiaw, M Maaouia / Eur. J. Pure Appl. Math, 13 (3) (2020), 472-482 481

Proof. ∗ For n = 0,
F = TorS

−1A
n (-, S−1B) = -⊗S−1AS

−1B and G = Êxt
n

S−1A(-, S−1B)o = HomS−1A(-,B)o.
By the proposition 4, the functors -⊗S−1AS

−1B = TorS
−1A

0 (-, S−1B) andHomS−1A(-,B)o =

Êxt
0

S−1A(-,B)o are adjoint.
So for n = 0 the property is verified.

∗ Suppose that the property is true up to the order n, i.e. the functors F = TorS
−1A

n (-,B) andG =

Êxt
n

S−1A(-,B)o are adjoint.

∗ Show that the functors F = TorS
−1A

n+1 (-, S−1B) and G = Êxt
n+1

S−1A(-, S−1B)o are ad-
joint.
Let

PS−1B : · · · −→ S−1Pn+1
dn+1−→ S−1Pn

dn−→ · · · −→ S−1P2
d2−→ S−1P1

d1−→ S−1P0
ε−→

S−1B −→ 0 be a projective resolution of S−1B.
Pose

K0 = Kerε and Kn = Kerdn, ∀ n ≥ 1.

By the proposition 3 we have:

TorS
−1A

n+1 (-, S−1B) and TorS
−1A

1 (-,Kn−1) are naturally isomorphic functors.

Êxt
n+1

S−1A(-, S−1B)o and Êxt
1

S−1A(-,Kn−1)
o are naturally isomorphic functors.

According to the inductive hypothesis, the functors TorS
−1A

1 (-,Kn−1) and Êxt
1

S−1A(-,Kn−1)
o

are adjoint.

So we have Êxt
1

S−1A(-,Kn−1)
o and TorS

−1A
1 (-,Kn−1) which adjoint, also TorS

−1A
1 (-,Kn−1)

and TorS
−1A

n+1 (-, S−1B) are naturally isomorphic functors, so by the proposition 2 the func-

tors Êxt
1

S−1A(-,Kn−1)
o and TorS

−1A
n+1 (-, S−1B) are adjoint.

So we have TorS
−1A

n+1 (-, S−1B) and Êxt
1

S−1A(-,Kn−1)
o which are adjoint, also Êxt

1

S−1A(-,Kn−1)
o

and Êxt
n+1

S−1A(-, S−1B)o are naturally isomorphic functors, so by the proposition 2 the

functors TorS
−1A

n+1 (-, S−1B) and Êxt
n+1

S−1A(-, S−1B)o are adjoint.

Hence, the functors TorS
−1A

n (-, S−1B) and Êxt
n

S−1A(-, S−1B)o are adjoint for all n ≥ 0 .

Corollary 4. Let A be a duo ring, P a prime ideal of A, S = (A-P ) ∩ Z(A) and B an
(A-A)-bialgebra. Then

TorS
−1A

n (-, S−1B) : Alg-S−1A� S−1A-Modo : Êxt
n

S−1A(-, S−1B)o

is an adjunction.



REFERENCES 482

Proof. Since A is a duo ring, then A-P is a multiplicatively closed subset of A, so
S = (A-P ) ∩ Z(A) is a central multiplicatively closed subset of A.

So by the previous theorem TorS
−1A

n (-, S−1B) is a left adjoint to Êxt
n

S−1A(-, S−1B)o.

Corollary 5. Let A be a duo ring, P a prime ideal of A, SR the set of regular elements
of A-P , S = SR ∩ Z(A) and B an (A-A)-bialgebra. Then

TorS
−1A

n (-, S−1B) : Alg-S−1A� S−1A-Modo : Êxt
n

S−1A(-, S−1B)o

is an adjunction.

Proof. Since A is a duo ring, then the set of regular elements SR of A-P is a multi-
plicatively closed subset of A, so S = SR ∩Z(A) is a central multiplicatively closed subset
of A.
So by the previous theorem TorS

−1A
n (-, S−1B) is a left adjoint to Êxt

n

S−1A(-, S−1B)o.

Acknowledgements

This work was supported by the UFR-SAT.

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