Clean coalgebras and clean comodules of finitely generated projective modules
Let R be a commutative ring with multiplicative identity and P is a finitely generated projective R-module. If P* is the set of R-module homomorphism from P to R, then the tensor product P* ⊗R P can be considered as an R-coalgebra. Furthermore, P and P* is a comodule over coalgebra P* ⊗R P. Using t...
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irk-123456789-1887102023-03-12T01:28:54Z Clean coalgebras and clean comodules of finitely generated projective modules Puspita, N.P. Wijayanti, I.E. Surodjo, B. Let R be a commutative ring with multiplicative identity and P is a finitely generated projective R-module. If P* is the set of R-module homomorphism from P to R, then the tensor product P* ⊗R P can be considered as an R-coalgebra. Furthermore, P and P* is a comodule over coalgebra P* ⊗R P. Using the Morita context, this paper give sufficient conditions of clean coalgebra P* ⊗R P and clean P* ⊗R P-comodule P and P*. These sufficient conditions are determined by the conditions of module P and ring R. 2021 Article Clean coalgebras and clean comodules of finitely generated projective modules / N.P. Puspita, I.E. Wijayanti, B. Surodjo // Algebra and Discrete Mathematics. — 2021. — Vol. 31, № 2. — С. 251–260. — Бібліогр.: 17 назв. — англ. 1726-3255 DOI:10.12958/adm1415 2020 MSC: 16T15, 16D90, 16D40. http://dspace.nbuv.gov.ua/handle/123456789/188710 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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Let R be a commutative ring with multiplicative identity and P is a finitely generated projective R-module. If P* is the set of R-module homomorphism from P to R, then the tensor product P* ⊗R P can be considered as an R-coalgebra. Furthermore, P and P* is a comodule over coalgebra P* ⊗R P. Using the Morita context, this paper give sufficient conditions of clean coalgebra P* ⊗R P and clean P* ⊗R P-comodule P and P*. These sufficient conditions are determined by the conditions of module P and ring R. |
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Puspita, N.P. Wijayanti, I.E. Surodjo, B. |
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Puspita, N.P. Wijayanti, I.E. Surodjo, B. Clean coalgebras and clean comodules of finitely generated projective modules Algebra and Discrete Mathematics |
| author_facet |
Puspita, N.P. Wijayanti, I.E. Surodjo, B. |
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Puspita, N.P. |
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Clean coalgebras and clean comodules of finitely generated projective modules |
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Clean coalgebras and clean comodules of finitely generated projective modules |
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Clean coalgebras and clean comodules of finitely generated projective modules |
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Clean coalgebras and clean comodules of finitely generated projective modules |
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Clean coalgebras and clean comodules of finitely generated projective modules |
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clean coalgebras and clean comodules of finitely generated projective modules |
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Інститут прикладної математики і механіки НАН України |
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2021 |
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http://dspace.nbuv.gov.ua/handle/123456789/188710 |
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Clean coalgebras and clean comodules of finitely generated projective modules / N.P. Puspita, I.E. Wijayanti, B. Surodjo // Algebra and Discrete Mathematics. — 2021. — Vol. 31, № 2. — С. 251–260. — Бібліогр.: 17 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
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AT puspitanp cleancoalgebrasandcleancomodulesoffinitelygeneratedprojectivemodules AT wijayantiie cleancoalgebrasandcleancomodulesoffinitelygeneratedprojectivemodules AT surodjob cleancoalgebrasandcleancomodulesoffinitelygeneratedprojectivemodules |
| first_indexed |
2025-07-16T10:53:47Z |
| last_indexed |
2025-07-16T10:53:47Z |
| _version_ |
1837800598009806848 |
| fulltext |
“adm-n2” — 2021/7/19 — 10:26 — page 251 — #87
© Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 31 (2021). Number 2, pp. 251–260
DOI:10.12958/adm1415
Clean coalgebras and clean comodules
of finitely generated projective modules∗
N. P. Puspita, I. E. Wijayanti, and B. Surodjo
Communicated by R. Wisbauer
Abstract. Let R be a commutative ring with multiplicative
identity and P is a finitely generated projective R-module. If P ∗ is
the set of R-module homomorphism from P to R, then the tensor
product P ∗⊗RP can be considered as an R-coalgebra. Furthermore,
P and P ∗ is a comodule over coalgebra P ∗ ⊗R P . Using the Morita
context, this paper give sufficient conditions of clean coalgebra
P ∗ ⊗R P and clean P ∗ ⊗R P -comodule P and P ∗. These sufficient
conditions are determined by the conditions of module P and ring R.
Introduction
In this paper a commutative ring with the identity is denoted by R.
A ring R is said to be a clean ring if every element of R can be express as
a sum of a unit and an idempotent element [1]. Moreover, a clean ring is
one of the subclasses of exchange rings [2, 3]. The previous authors have
given some notions of clean rings and exchange rings for example [4–9].
Some authors have been studied the endomorphism structure of R-
modules M . It is proved that the ring of a linear transformation of a
countable linear vector space is clean [10] and the result is also true for
arbitrary vector spaces over a field and any vector space over a division
∗This research is supported by Doctoral Research Grant, Directory of Higher Edu-
cation, the Ministry of Education and Culture, Indonesia, 2019-2020.
2020 MSC: 16T15, 16D90, 16D40.
Key words and phrases: clean coalgebra, clean comodule, finitely generated
projective module, Morita context.
“adm-n2” — 2021/7/19 — 10:26 — page 252 — #88
252 Clean coalgebras and clean comodules of FGP
ring, it is has been proved in [11] and [12]. An R-module M is called a clean
module if EndR(M) is a clean ring [13]. We recall the important result
of [13], i.e., necessary and sufficient conditions of clean elements in an
endomorphism ring (see Proposition 2.2 and Proposition 2.3). Furthermore
in [14] the authors prove this property in a shorter way by proving that
every non M -singular self-injective module M is clean (see Lemma 4).
The structure of comodules and coalgebras has been introduced in
1969 by Sweedler. He introduced a coalgebra over a field as the dualization
of algebras over a field. Later, this ground field has been generalized to
any ring with multiplicative identity [15]. Furthermore, a comodule over
a coalgebra is well-known as a dualization of a module over a ring. For
any R-coalgebra C we can construct C∗ = HomR(C,R), where C∗ is an
algebra (ring) over convolution product. We called C∗ as a dual algebra
of C. Hence, we have an important result, i.e. if M is a right C-comodule,
then M is a left module over the dual algebra C∗. Moreover, for any
M,N ∈ M
C and EndC(M,N) ⊆C∗ End(M). Thus, the category of right
C-comodule (MC) is a subcategory of left C∗-module (C∗M
C). In [15],
the R-coalgebra C satisfies the α-condition if and only if MC is a full
subcategory of C∗M
C. Moreover, the EndC(M) =C∗ End(M) if and only
if C is locally projective as an R-module (see [15]).
Recall the structure of comodules and coalgebras [15]. We applied the
notions of clean modules to comodule and coalgebra, and introduced the
following definition.
Definition 1. Let R be a ring and (C,∆, ε) an R-coalgebra. A right (left)
C-comodule M is called a clean comodule if the endomorphism ring of
right (or left) C-comodule M (denoted by EndC(M) (or C End(M)) is a
clean ring.
Definition 1 means that if C satisfies the α-condition, the right C-
comodule M is a clean comodule if and only if the ring C∗ End(M) is a
clean ring, since EndC(M) =C∗ End(M). Since every R-coalgebra C is a
right and left comodule over itself, based on Definition 1 we introduce a
clean coalgebra.
Definition 2. Let R be a ring. An R-colagebra C is called a clean
coalgebra if C is a clean comodule over itself.
If C satisfies the α-condition, Definition 2 means that C is a clean
coalgebra if C is clean as a C∗-module. We present the trivial of clean
coalgebra. Consider any ring R as an R-coalgebra with the trivial co-
multiplication ∆T : R → R ⊗R R, r 7→ r ⊗ r, and counit εT : R → R,
“adm-n2” — 2021/7/19 — 10:26 — page 253 — #89
N. P. Puspita, I. E. Wijayanti, B. Surodjo 253
r 7→ r, for any r ∈ R. Hence, the dual algebra of R, i.e., (R∗,+, ∗) where
R∗ = EndR(R) is isomorphic to the ring R by mapping f 7→ f(1) for all
f ∈ R∗. Then we have a trivial R-coalgebra (R,∆T , εT ) is a clean if and
only if R is a clean ring.
Furthermore, since every ring R can be considered as the trivial R-
coalgebra (R,∆T , εT ), any R-module M is a (right and left)comodule over
coalgebra (R,∆T , εT ) with coaction
̺M : M 7→ M ⊗R R,m 7→ m⊗ 1.
It implies any R-module M is clean if and only if (M,̺M ) is a clean (right
and left) R-comodule, since R ≃ R∗.
Throughout P is a finitely generated (f.g) projective R-module and
P ∗ is a set of all R-module homomorphism from P to R. In [15] we
have already know that for any f.g projective module P and P ∗, we can
construct tensor product of P and P ∗, i.e., P ∗⊗RP . An R-module P ∗⊗RP
is an R-coalgebra by a comultiplication ∆ and counit ε as below:
Lemma 1. [15] Let P be a finitely generated projective R-module with
dual basis p1, p2, ..., pn ∈ P and π1, π2, ..., πn ∈ P ∗. The R-module P ∗⊗RP
is an R-coalgebra with the comultiplication and counit defined by
∆: P ∗ ⊗R P → (P ∗ ⊗R P )⊗R (P ∗ ⊗R P );
f ⊗ p 7→ Σif ⊗ pi ⊗ πi ⊗ p
and
ε : P ∗ ⊗R P → R, f ⊗ p 7→ f(p).
By the properties of the dual basis,
(IP ∗⊗RP ⊗∆)ε(f ⊗ p) = Σif ⊗ piπi(p) = f ⊗ p,
that is ε is a counit and the coassociativity of ∆ is proved by the following
equality
(IP ∗⊗RP ⊗∆)∆(f ⊗ p) = Σi,jf ⊗ pi ⊗ πi ⊗ pj ⊗ πj ⊗ p
= (∆⊗ IP ∗⊗RP )∆(f ⊗ p).
Furthermore, consider P and P ∗ as an R-module, then P and P ∗
respectively can be consider as a right and left comodule over R-coalgebra
P ∗ ⊗R P . By using the Morita context, which is we refer to [16], in this
paper we investigate the sufficient conditions of clean R-coalgebra P ∗⊗RP
“adm-n2” — 2021/7/19 — 10:26 — page 254 — #90
254 Clean coalgebras and clean comodules of FGP
and the cleanness of P and P ∗ as a P ∗⊗R P -comodule. In Morita context
we already know there are relationship between the structure of P, P ∗, R
and S = EndR(P ) [16]. The following theorem explain the relationship
between P and its dual, in which it is important to prove our main result.
Theorem 1. [16] Let R be a ring, P be a right R-module, S = EndR(P )
and Q = P ∗ = HomR(P,R). If P is a generator in R-MOD, then
1) α : Q⊗S P → R is an (R,R)-isomorphism;
2) Q ≃ HomS(SP,S S) as (R,S)-bimodules;
3) P ≃ HomS(QS , SS) as (S,R)-bimodules;
4) R ≃ End(SP ) ≃ End(QS) as rings.
Theorem 2. [16] Let R be a ring, P be a right R-module, S = EndR(P )
and Q = P ∗ = HomR(P,R). If P is finitely generated projective in
R-MOD, then
1) β : P ⊗R Q → S is an (S, S)-isomorphism;
2) Q ≃ HomR(PR, RR) as (R,S)-bimodules;
3) P ≃ HomR(RQ,R R) as (S,R)-bimodules;
4) S ≃ End(PR) ≃ End(RQ) as rings.
Since the cleanness of coalgebra and comodule are determined by
the structure of its endomorphism, using Theorem 2 and Theorem 1, we
observe when End(P ∗⊗RP )∗(P
∗ ⊗R P ),End(P ∗⊗RP )∗(P ) and
End(P ∗⊗RP )∗(P
∗) are clean.
1. The clean R-coalgebra P
∗ ⊗R P
Let P be an R-module. Here, we can construct tensor product of P
and P ∗. Furthermore, since R is a commutative ring, P ∗⊗R P ∼= P ⊗R P ∗
as an R-module. In this section we give some results which are related to
some conditions when the R-coalgebra P ∗⊗RP is clean. Let P be a finitely
generated projective R-module with basis p1, p2, ..., pn ∈ P and dual basis
π1, π2, ..., πn ∈ P ∗. Based on Theorem 2 we have P ⊗R P ∗ ∼= EndR(P ) as
an (S, S)-bimodule where
P ⊗R P ∗ → EndR(P ), p⊗ f 7→ [a 7→ pf(a)].
Now, consider the R-module P ∗ ⊗R P as an R-coalgebra, using the
Morita Context we have the following proposition.
Theorem 3. Let P be a finitely generated projective R-module with dual
basis p1, p2, ..., pn ∈ P π1, π2, ..., πn ∈ P ∗. If P is a clean R-module, then
the R-coalgebra P ∗ ⊗R P is clean.
“adm-n2” — 2021/7/19 — 10:26 — page 255 — #91
N. P. Puspita, I. E. Wijayanti, B. Surodjo 255
Proof. Let P be a finitely generated R-module and P ∗ = HomR(P,R)
is an R-module. Suppose that P is a clean R-module. Since P and R
is a finitely generated projective R-module, P ∗ = HomR(P,R) is also
a finitely generated projective R-module [17]. Here, we need to prove
weather R-coalgebra P ∗ ⊗R P satisfies the α-condition by proving the
tensor product of P ∗ ⊗R P is a projective R-module.
To show that P ∗ ⊗R P is projective as an R-module, we must show
that for any surjective map f : A → B of R-module, the map
f∗ : HomR(P
∗ ⊗R P,A) → HomR(P
∗ ⊗R P,B)
is also surjective. Since P ∗ is a projective R-module so that
h : HomR(P
∗, A) → HomR(P
∗, B)
is surjective. By projectivity of P we obtain
h∗ : HomR(P,HomR(P
∗, A)) → HomR(P,HomR(P
∗, B))
is also surjective. Put C = A or B, then by [17] (see page 425) we have
HomR(P,HomR(P
∗, C)) ≃ HomR(P
∗ ⊗R P,C)
It implies that f is isomorphic to h∗, and moreover f is a surjective map.
Thus, P ∗ ⊗R P is a projective R-module. Therefore as an R-coalgebra,
P ∗ ⊗R P satisfies the α-condition. Then we have
(P ∗⊗RP )∗ End(P
∗ ⊗R P ) ≃ (P ∗ ⊗R P )∗.
We are going to show that P ∗ ⊗R P is a clean R-coalgebra, it means we
need to prove that (P ∗ ⊗R P )∗ is a clean ring (see Proposition 4.1.8).
Based on [17], we have a relationship between tensor product and R-
module homomorphism. Furthermore, since P is finitely generated, the
dual algebra P ∗ ⊗R P is isomorphic to the ring EndR(P ) by the bijective
map as below:
(P ∗ ⊗R P )∗ = HomR(P
∗ ⊗R P,R) ≃ HomR(P,HomR(P
∗, R))
≃ HomR(P, P
∗∗) ≃ EndR(P )(since P ∗∗ ≃ P ).
Hence, if P is a clean R-module, then EndR P is a clean ring. It
means (P ∗ ⊗R P )∗ ≃ EndR(P ) is a clean ring. Since P ∗ ⊗R P ∗ ≃
End(P ∗⊗RP )∗(P
∗⊗RP ) is a clean ring, P ∗⊗RP is a clean R-coalgebra.
“adm-n2” — 2021/7/19 — 10:26 — page 256 — #92
256 Clean coalgebras and clean comodules of FGP
For P = R we obtain P ∗ = R∗ = EndR(R) ≃ R and R∗ ⊗R R ≃ R is
a coassociative R-coalgebra with counital. Thus, if R is a clean R-module
(i.e., R is clean as a ring), then (R,∆, ε) is a clean coalgebra over itself.
Recall the example of R-coalgebra Mn(R) (see [15]). The matrix ring
Mn(R) is an R-coalgebra by the coproduct and counit as below
∆ : Mn(R) → Mn(R)⊗R Mn(R), eij 7→ Σi,jei,k ⊗ ekj , (1)
and
ε : Mn(R) → R, eij 7→ δi,j . (2)
It is called the (n, n)-matrix coalgebra over R. Throughout, the R-
coalgebra Mn(R) with the comultiplication (1) and the counit (2) denoted
by MC
n (R). Furthermore, we will show that the R-coalgebra MC
n (R) can
be identified as an R-coalgebra P ∗ ⊗R P when P = Rn.
Lemma 2. Let P = Rn. Then the comultiplication and counit on R-
coalgebra (Rn)∗ ⊗R Rn is equivalent to the comultiplication and counit of
R-coalgebra MC
n (R). It means (Rn)∗ ⊗R Rn ≈ MC
n (R) .
Proof. Suppose that the canonical basis of Rn is {(0, 0, .., 1i, 0, ..0)}i∈N
and basis of (Rn)∗ is {πi}i∈N where πi((0, 0, .., 1j , 0, ..0)) = 1 for i = j and
0 for i 6= j. Therefore
1) The comultiplication
∆ : (Rn)∗ ⊗R Rn → ((Rn)∗ ⊗R Rn)⊗R (Rn)∗ ⊗R Rn
f ⊗ p 7→
∑
i f ⊗ pi ⊗ πi ⊗ p
For any f =
∑
i aiπi and p =
∑
i bipi ∈ Rn we have
∆(f ⊗ p) =
∑
k
(
∑
i
aiπi)⊗ pk ⊗ πk ⊗ (
∑
j
bjpj)
=
∑
k
(
∑
i
aiπi(pk))⊗ (
∑
j
bjπk(pj))
Since (Rn)∗ ⊗R Rn ≈ Mn(R) as an R-module by mapping πi ⊗ pj 7→ eij
for any i, j, we have
∆(f ⊗ p) ≃
∑
k
(
∑
i
aieik ⊗
∑
j
bjekj).
“adm-n2” — 2021/7/19 — 10:26 — page 257 — #93
N. P. Puspita, I. E. Wijayanti, B. Surodjo 257
It implies the case f ⊗ p = πi ⊗ pj ≈ eij , we have
∆(f ⊗ p) = ∆(eij) = ∆(πi ⊗ pj) =
∑
k
πi ⊗ pk ⊗ πk ⊗ pj
=
∑
k
πi(pk)⊗ πk(pj) =
∑
k
eik ⊗ ekj
Therefore,
∆(πi ⊗ pj) ≈ ∆(eij) =
∑
k
eik ⊗ ekj .
Consequently, this result similar to the comultiplication on R-coalgebra
MC
n (R).
2) The counit of (Rn)∗ ⊗R Rn is ε(f ⊗ p) = f(p). For any f ⊗ p ∈
(Rn)∗ ⊗R Rn where f =
∑
i aiπ and p =
∑
j bjpj ∈ Rn we have
ε(f ⊗ p) = ε(
∑
i
aiπi ⊗
∑
j
bjpj) =
∑
i
aiπi(
∑
j
bjpj)
= ε(f ⊗ p) =
∑
i
ai
∑
j
bjπi(pj) = aibi.
Related with an R-coalgebra MC
n (R), for canonical basis eij ≈ πi ⊗ pj
(see Lemma 2). Putting f ⊗ p = πi ⊗ pj ∈ (Rn)∗ ⊗R Rn (see Lemma 2),
then
ε(eij) = ε(πi ⊗ pj) = πi(pj) = δi,j .
It is analogue to the counit of MC
n (R).
It is clear that every ring is a trivial coalgebra over itself ([15]). On the
other hand, we have already known that a ring R is clean if and only if
(R,∆T , εT ) is a clean R-coalgebra. Furthermore, if R is a clean ring, then
the ring Mn(R) is a clean ring [1]. Now, let consider the matrix ring Mn(R)
as a coalgebra over itself by the trivial comultiplication (∆T )and counit
(εT ), denoted by (Mn(R),∆T , εT ). Hence, if Mn(R) is a clean ring, then
(Mn(R),∆T , εT ) is a clean coalgebra over itself. The following corollary
explains the cleanness of R-coalgebra MC
n (R) with ∆ and ε in Equation
(1) and (2).
Corollary 1. If R is a clean ring, then the R-coalgebra MC
n (R) is clean.
Proof. By Lemma 2 MC
n (R) is a special case of P ∗ ⊗R P when P = Rn.
Suppose that P = Rn. Since R is a clean ring, then Rn is a clean R-module
[13]. By the Theorem 3 (Rn)∗ ⊗R Rn = MC
n (R) is a clean R-coalgebra,
since Rn is a clean R-module.
“adm-n2” — 2021/7/19 — 10:26 — page 258 — #94
258 Clean coalgebras and clean comodules of FGP
2. The cleanness of P and P
∗ as a P
∗ ⊗R P -comodule
Let R be a commutative ring with multiplicative identity and P be a
finitely generated projective R-module. In [15] if P is a clean R-module,
then the R-coalgebra P ∗⊗RP is clean. If P is a finitely generated projective
R-module with basis p1, p2, ..., pn ∈ P and dual basis π1, π2, ..., πn ∈ P ∗,
then P is a right P ∗ ⊗R P -comodule with the coaction
̺P : P → P ⊗R (P ∗ ⊗R P ), p 7→ Σipi ⊗ πi ⊗ p.
P is a subgenerator in M
P ∗⊗RP and there is a category isomorphism
M
P ∗⊗RP ≃ MEndR(P ).
The dual P ∗ is a left P ∗ ⊗R P -comodule with the coaction
P ∗
̺ : P ∗ → (P ∗ ⊗R P )⊗R P ∗, f 7→ Σif ⊗ pi ⊗ πi.
Here, we will investigate the conditions under which P and P ∗ are
clean comodules over P ∗ ⊗R P .
Theorem 4. Let P be a finitely generated projective R-module with basis
p1, p2, ..., pn ∈ P and dual basis π1, π2, ..., πn ∈ P ∗. If R is a clean ring,
then P is a right clean P ∗⊗R P -comodule and P ∗ is a left clean P ∗⊗R P -
comodule.
Proof. 1) Suppose that P is a projective R-module. Consider P as a right
P ∗ ⊗R P -comodule. We want to prove that P is a right clean P ∗ ⊗R P -
comodule, i.e., (P ∗⊗RP )∗ End(P ) is a clean ring.
Based on [15], since P ∗ ⊗R P is a finitely generated projective R-
module, R-coalgebra P ∗ ⊗R P satisfies the α-condition and we have the
following condition:
(P ∗⊗RP )∗M ≃ M
P ∗⊗RP . (3)
On the other hand, it is true that the ring (P ∗ ⊗R P )∗ ≃ EndR(P ).
Therefore,
M
P ∗⊗RP ≃(P ∗⊗RP )∗ M ≃EndR(P ) M.
We are going to prove that the ring (P ∗⊗RP )∗ End(P ) ∈(P ∗⊗RP )∗ M is
clean. Based on Equation (3) and using the Morita Context (see Theorem
1), since P is a generator, R ≃ End(EndR(P )P ) as a ring. Therefore,
(P ∗⊗RP )∗ End(P ) ≃ EndEndR(P )(P ) and EndEndR(P )(P ) ≃ R
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N. P. Puspita, I. E. Wijayanti, B. Surodjo 259
as a ring. Hence, (P ∗⊗RP )∗ End(P ) ≃ R as an R-module. Noted that R is
a clean ring if and only if R is a clean R-module. Then
(P ∗⊗RP )∗ End(P ) ≃ R
is a clean ring. Consequently, P is a clean P ∗ ⊗R P -comodule.
2) Consider P ∗ as a left P ∗ ⊗R P -comodule. We want to prove that
P ∗ is a left P ∗ ⊗R P -comodule, i.e., End(P ∗⊗RP )∗(P
∗) is a clean ring.
Analogue with point (1) we have
P ∗⊗RP
M ≃ M(P ∗⊗RP )∗ ≃ MEndR(P ).
We going to prove that the ring End(P ∗⊗RP )∗(P
∗) ∈ M(P ∗⊗RP )∗ is clean.
From Equation (3) we have
End(P ∗⊗RP )∗(P
∗) ≃ EndEndR P (P
∗).
Furthermore, from Theorem 1 we have R ≃ EndEndR(P )(P
∗).
Therefore,
End(P ∗⊗RP )∗(P
∗) ≃ R.
Consequently, if R is a clean ring then (P ∗⊗RP )∗ End(P ) ≃ R is a clean
ring. Hence, P ∗ is a left clean P ∗ ⊗R P -comodule.
Remark 1. Let P = Rn. As a special case for any n ∈ N then Rn is a
comodule over the coalgebra MC
n (R). On the other hand, if R is a clean
ring (i.e., a clean R-module), then Rn is a clean R-module. Therefore, if
R is a clean R-module, then Rn is a right clean MC
n (R)-comodule.
This paper gives the sufficient conditions of clean R-coalgebra P ∗⊗RP
and the cleanness of P and P ∗ as a P ∗ ⊗R P -comodule. We already get
some conclusions i.e., if P is a clean R-module, then the R-coalgebra
P ∗ ⊗R P is clean and if R is a clean ring, then P (resp. P ∗) is a right
(resp. left) clean P ∗⊗RP -comodule. We see that the cleanness of P ∗⊗RP
depends on R if P is a finitely generated projective R-module (i.e., it is
very closed to free R-module).
References
[1] Nicholson, W.K., Lifting Idempotents and Exchange Rings, Trans. Amer. Math.
Soc., 229, 1977, 269–278.
[2] Warfield, Jr., R. B., Exchange rings and decompositions of modules, Math. Ann.
199, 1972, 31–36.
[3] Crawley, P., and Jónnson, B., Refinements for Infinite Direct Decompositions
Algebraic System, Pacific J. Math., 14, 1964, 797–855.
“adm-n2” — 2021/7/19 — 10:26 — page 260 — #96
260 Clean coalgebras and clean comodules of FGP
[4] Camillo, V.P., and Yu, H.P., Exchange Rings, Units and Idempotents, Comm.
Algebra, 22(12), 1994, 4737–4749.
[5] Han, J., and Nicholson, W.K., Extension of Clean Rings, Comm. Algebra, 29(6),
2001, 2589–2595.
[6] Anderson, D.D., and Camillo, V.P., Commutative Rings Whose Element are a
sum of a Unit and Idempotent, Comm. Algebra, 30(7), 2002, 3327–3336.
[7] Tousi, M., and Yassemi, S., Tensor Product of Clean Rings, Glasgow Math. J, 47.
2005, 501–503.
[8] McGovern, W. Wm., Characterization of commutative clean rings, Int. J. Math.
Game Theory Algebra, 15(40), 2006, 403–413.
[9] Chen, H., and Chen, M., On Clean Ideals, IJMMS, 62, 2002, 3949–3956.
[10] Nicholson, W. K., and Varadarajan, K., Countable Linear Transformations are
Clean, Proceedings of American Mathematical Socienty, 126, 1998, 61–64.
[11] Scarcoid, M.O., Perturbation the Linear Transformation By Idempotent, Irish
Math. Soc. Bull., 39, 1997, 10–13.
[12] Nicholson, W.K., Varadarajan, K. and Zhou, Y., Clean Endomorphism Rings,
Archiv der Mathematik, 83, 2004, 340–343.
[13] Camillo, V.P., Khurana, D., Lam, T.Y., Nicholson, W.K. and Zhou, Y., Continous
Modules are Clean, J. Algebra, 304, 2006, 94–111.
[14] Camillo, V.P., Khurana, D., Lam, T.Y., Nicholson, W.K. and Zhou, Y., A Short
Proof that Continous Modules are Clean, Contemporary Ring Theory 2011, Pro-
ceedings of the Sixth China-Japan-Korea International Conference on Ring Theory,
2012 165–169.
[15] Brzeziński, T., and Wisbauer, R., Corings and Comodules, Cambridge University
Press, United Kingdom, 2003.
[16] Lam, T.Y., Graduated Texts in Mathematics: Lectures on Modules and Rings,
Springer-Verlag, New York, inc, 1994.
[17] Adkins, W.A., and Weintraub, S. H., Algebra "An Approach via Module Theory",
Springer-Verlag New York, Inc., USA, 1992.
Contact information
Nikken Prima
Puspita,
Indah Emilia
Wijayanti,
Budi Surodjo
Department of Mathematics, Faculty of
Mathematics and Natural Science, Universitas
Gadjah Mada, Yogyakarta, Indonesia
E-Mail(s): nikken.prima.p@mail.ugm.ac.id
(nikkenprima@gmail.com),
ind_wijayanti@ugm.ac.id,
surodjo_b@ugm.ac.id
Web-page(s): www.indahwijayanti.staff.
ugm.ac.id,
www.acadstaff.ugm.ac.id/
MTk2NTExMjYxOTkxMDMxMDAx
Received by the editors: 10.07.2019
and in final form 23.10.2020.
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