Graded Modules as a Clean Comodule

In ring and module theory, the cleanness property is well established. If any element of R can be expressed as the sum of an idempotent and a unit, then R is said to be a clean ring. Moreover, an R-module M is clean if the endomorphism ring of M is clean. We study the cleanness concept of coalgebra and comodules as a dualization of the cleanness in rings and modules. Let C be an R-coalgebra and M be a C-comodule. Since the endomorphism of C-comodule M is a ring, M is called a clean C-comodule if the ring of C-comodule endomorphisms of M is clean. In Brzeziński and Wisbauer (2003), the group ring R[G] is an R-coalgebra. Consider M as an R[G]-comodule. In this paper, we have investigated some sufficient conditions to make M a clean R[G]-comodule, and have shown that every G-graded module M is a clean R[G]-comodule if M is a clean R-module.


Introduction
Throughout, R is a commutative ring with identity, G is a group and M is an R-module. Research into clean rings is an interesting topic in algebra. In their 1977 paper, Nicholson introduced the concept of a clean element. Any element x of R is said to be clean if x = u + e with e is an idempotent element and u is a unit element in R. Moreover, if its elements are clean, then R is a clean ring. Some authors, including R. B. Warfield, Jr. (1972) and Crawley and Jónnson (1964), studied and proved certain properties of exchange rings. Clean rings are a subclass of exchange rings. Another result of clean rings has already been by Han and Nicholson (2001) and Anderson and Camillo (2002). Consider the group ring R [G]. It is possible to study the clean element in R[G] as a ring. Hence, Han and Nicholson (2001) and McGovern (2006) give the necessary and sufficient conditions for the group ring R[G] to be clean.
Consider R as an R-module over itself. The endomorphisms of R-module R are denoted by (End R (R)). Hence, the ring of End R (R) is isomorphic to R. This implies that if R is clean, then so is End R (R). The converse is also true. This condition is generalized for any R-module M. For any R-module M, we can construct the endomorphism ring of M over R, denoted by End R (M). The cleanness of M is determined by the structure of End R (M). An R-module M is clean if the ring of End R (M) is clean. Camillo et al. (2006) give the conditions that make the ring of End R (M) clean. As the main result, the paper proves that continuous modules are clean. Now, recall the notion of comodules over a coalgebra over a field as described by Sweedler (1969). Brzeziński and Wisbauer (2003) give the generalized results for Sweedler by generalizing the ground field to any commutative ring with identity.
Throughout this paper, (C, ∆, ε) is a coassociative and counital coalgebra over R. Hence, the comultiplication ∆ : C → C ⊗ R C and the counit ε : C → R are R-module homomorphisms which satisfy some conditions (see (Brzeziński and Wisbauer, 2003)). The R-module M with R-module homomorphism M : M → M ⊗ R C, which is called a right Ccoaction, with the properties as below (coassociative and counital): is called a right C-comodule. In this paper, the notation (M, M ) is a right C-comodule M with C-right coaction M . In module theory, we are already familiar with homomorphism between two modules. In the comodule theory, we have a similar concept to module homomorphism, i.e., the comodule morphism. Let (M, M ) and (N, N ) be right C-comodule.
with I C is the identity map of C. The set of all right C-comodule morphisms from M to N is denoted by Hom C (M, N). Moreover, the category of C-comodules is denoted by M C . If M = N, then the endomorphism of C-comodule M, i.e., End C (M) is a ring over the addition and composition product.
From the R-coalgebra C we can construct the dual R-algebra C * = Hom R (C, R) by defining the convolution product as below: for any f 1 , f 2 ∈ C * , µ is the multiplication of R and ∆ is the comultiplication of R-coalgebra C. In Brzeziński and Wisbauer (2003), (C * , +, * ) is a dual R-algebra. Moreover, any right C-comodule M can be considering as a left C * -module by the following scalar multiplication : By Sweedler's notation, for any g ⊗ a ∈ C * ⊗ R M, (I M ⊗ g) • M (a) = a 0 g(a 1 ) ∈ M. On the other hand, every right C-comodule morphism is a left C * -module morphism such that End C (M) is a subring of C * End(M). The relation between them is isomorphic if and only if C is locally projective as an R-module (see the α-condition (Brzeziński and Wisbauer, 2003)). Thus, if C is a locally projective R-module, End C (M) C * End(M). In this paper, the notion of clean comodules depends on the structure of End C (M).
The graded ring R[G] is an R-coalgebra, by taking the comultiplication ∆ : When we have graded modules in comodules, it can be considered a comodule over the coalgebra R[G]. Our paper will focus on graded modules as the comodules structure. We recall the definition of graded rings and graded modules, which are referred to by Natasescu and Oystaeyen in their 1982 and 2004 papers.
Definition 1.1 (Natasescu and Oystaeyen, 2004) Let G be a group (or semigroup) with identity element e. A ring R is called a G-graded ring if there is a family of additive subgroups of R i.e., For any group G, the category of G-graded rings is denoted by G − RING. The objects of G − RING are the class of G-graded rings and the set of all graded homomorphisms. If G = {e}, then G − RING and the category of rings coincide. Definition 1.3 (Natasescu and Oystaeyen, 1982) Let G be a group and R = ⊕ σ∈G R σ a graded ring of type G. Let M and N be G-graded modules over R. An R-module homomorphism f : Let R be a G-graded ring. The category of G-graded (left) modules over R is denoted by gr M. The set of all morphisms from M to N in the category of gr M is denoted by Hom gr (M, N). Hence, the category gr M is a subcategory of R-modules R M and the set of graded endomorphisms of M, denoted by End gr (M), is a subring of End R (M). Let G be a group and (R[G], ∆ 1 , ε 1 ) be an R-coalgebra with comultiplication and counit as defined in Brzeziński and Wisbauer (2003). We have the following relationship: Theorem 1.4 (Brzeziński and Wisbauer, 2003) Let G be a group (semigroup). Considering a ring R with the trivial grading, an R-module M is G-graded if and only if it is an R[G]-comodule.
Theorem 1.4 means that when we have an arbitrary graded module over a ring R with type G, we always construct a comodule over coalgebra R [G]. In this paper, we will focus on this fact. For any (right) C-comodule M, the comodule endomorphism (End C (M), +, •) is a ring with addition and composition operations. We are going to use this fact to define the notion of a clean comodule. On the other hand, every graded module over R with type G is an R[G]-comodule. Using some properties of comodule categories, module categories, and graded modules, we obtain a condition that makes R[G]comodules clean comodules. Our main result is that if M is a clean R-module, then the G-graded module M over R is a clean comodule over an R-coalgebra R[G] .

Results
The concept of clean comodules has become to the concept of clean modules. Whether any R-module M is clean or not clean depends on the structure of the endomorphism ring of End R (M). Hence, for any M ∈ M C , we also have the fact that the endomorphism of C-comodule, i.e., (End C (M), +, •) is a ring with unity. This fact is used to define a clean concept in a comodule structure. Based on the definition of clean modules, we introduce clean comodules.
Definition 2.1 A C-comodule M is called a clean C-comodule if the endomorphism ring of End C (M) is clean The clean coalgebra is defined as a special case of clean comodules. Since any R-coalgebra C is a C-comodule, C is said to be clean if C is clean as a comodule over itself or the endomorphism ring of End C (C) is clean. In (Brzeziński and Wisbauer, 2003), if C is a locally projective module, then we have a special condition related to the relationship between End C (M) and C * End(M). Hence, C * End(M) End C (M), Moreover, for R-coalgebra C, C is clean if and only if Any ring R can be considered a trivial coalgebra by comultiplication ∆ : R → R⊗ R R, r → r⊗r and counit ε : R → R, r → 1. Hence, the ring End R (R, R) R by mapping f → f (1). Consequently, if R is a clean ring, then R is also clean as an Rcoalgebra. For another example, we offer the following case.
Example 2.2 Let the ring Z 5 . Consider Z 5 a Z 5 -coalgebra by the (coassociative) comultiplication as below: Since Z 5 is a field, then it is a locally projective module over itself. Consequently, the End Z 5 (Z 5 ) Z * 5 End(Z 5 ) Z * 5 (Brzeziński and Wisbauer, 2003). Hence, Z * 5 = End Z 5 (Z 5 ) = {g 0 , g 1 , g 2 , g 3 , g 4 } where g 0 is a zero map, g 1 is an identity map of Z 5 . Otherwise, As a dual R-algebra and by using the convolution product of * , i.e., for any g i , g j ∈ Z * 5 and i, j = 1, 2, 3, 4, 5. We have the following table. Table 2.1. The Result of Convolution Product in Z * 5 * g 0 g 1 g 2 g 3 g 4 g 0 g 0 g 0 g 0 g 0 g 0 g 1 g 0 g 2 g 4 g 1 g 3 g 2 g 0 g 1 g 3 g 2 g 4 g 3 g 0 g 1 g 2 g 3 g 4 g 4 g 0 g 3 g 1 g 4 g 2 Based on the result of Table 2.1, we can see that the multiplicative identity of Z * 5 is g 3 and any non-zero element of Z * 5 is a unit. Thus, the ring Z * 5 is a field. Clearly, any field is a clean ring. This means that the ring End Z 5 (Z 5 ) which is isomorphic with the ring Z * 5 is clean. This fact implies that the Z 5 -coalgebra (Z 5 , ∆, ε) is clean. In this paper we will focus on investigating the cleanness of comodule M over R-coalgebra R [G]. That is, we will concentrate on proving the endomorphism ring End R [G] (M) to be clean. Throughout, ring R is a G-graded ring, M R[G] is the category of R[G]-comodules. Defining the following map we obtain an equivalence functor as we find in the following Lemma 2.3 Let G be a group and R be a G-graded ring by trivial grading. The map F in Equation (8)  For any m = g∈G m g ∈ M we have Hence, for any f in Hom gr (M, N) we have F( f ) in Hom R[G] (M, N). In particular, for any G-graded homomorphism f we can consider this f as a C-comodule morphism. Thus, we conclude that the functor F is well-defined. Moreover, it is easy to prove that F is a covariant functor.
Furthermore, we prove that F is an equivalence functor. For any M, N in gr M, Since for all g ∈ G, β(m g ) = n g , β is a G-graded homomorphism. Therefore, for any β ∈ Hom R[G] (M, N), there exist β ∈ Hom gr (M, N), such that F(β) = β. Thus F is full.
Based on Theorem 1.4, functor F is essentially surjective. Hence, F is a equivalence functor and moreover, the category . In general, a G-graded module over R is not necessarily a clean R-module. As an example, Z is a Z-graded module by trivial grading. However, we know that Z is not a clean module over itself. Based on these reasons and some related properties between R M, gr M and M R[G] , we obtain the following theorem.
Theorem 2.5 Let R be a ring, G be a group and M be a clean R-module. Consider R as a G-graded ring by trivial grading. If M is a G-graded module over R, then M is a clean R[G]-comodule.
Proof Based on Brzeziński and Wisbauer (2003), M is an R[G]-comodule. Since R is a G-graded ring by trivial grading, R e = R and R g = 0 for all g e. This implies gr M Re M = R M.
For the first case, we need to prove that End gr (M) is also a clean ring. Take any f ∈ End gr (M) ⊆ End R (M). Since M is a clean R-module, f = u + e for a unit u and an idempotent e in End R (M). Recall the functor R ⊗ R e − with R e = R. For any morphism f : M → N in R M we get: in End gr (M) such that for any g∈G r g ⊗ R m ∈ gr M, we have Hence, e = I R ⊗ R e is an idempotent element in End gr (M).
From point (1) and (2), for any f in End gr (M) ⊆ End R (M), where f = u + e in End R (M), there exist u = I R ⊗ u a unit in End gr (M) and e = I R ⊗ e an idempotent element in End gr (M) such that f ≈ f = I R ⊗ R f = I R ⊗ R u + I R ⊗ R e = u + e , or f ≈ f is a clean element in End gr (M). Consequently, End gr (M) is a clean ring. This means that if the End R (M) is a clean ring, then End gr (M) is also a clean ring. Our objective is to prove that the End R[G] (M) is a clean ring. We derive the conclusion from previous facts. Based on Lemma , we have End R[G] (M) gr End(M).
As we have assumed, since M is a clean R-module, End R (M) is a clean ring. This fact implies the ring gr End(M) is also clean. Moreover, by isomorphisms in Equation 9 and clean properties in Lemma 2.4, we have the result that the endomorphism ring End R[G] (M) is clean. Consequently, M is a clean R[G]-comodule.
Example 2.6 Let G be a group and M an injective R-module. If R is a G-graded ring by trivial grading, then M is a G-graded R-module; moreover, M is a clean R[G]-comodule since any injective R-module is clean (Camillo et al., 2006).
The converse of Theorem 2.5 is also true. By functor F in Equation (8) if M is an R[G]-comodule, then M is a G-graded module over R. Therefore, we have the following theorem. Theorem 2.7 Let R be a ring, G a group. Consider R as a G-graded ring by trivial grading. If M is a clean R[G]-comodule, then M is a G-graded R-module and clean as an R-module.