On (weakly) precious rings associated to central polynomials

Let R be an associative ring with identity and let g(x) be a fixed polynomial over the center of R. We define R to be (weakly) g(x)-precious if for every element a∈R, there are a zero s of g(x), a unit u and a nilpotent b such that (a=±s+u+b) a=s+u+b. In this paper, we investigate many examples and properties of (weakly) g(x)-precious rings. If a and b are in the center of R with b-a is a unit, we give a characterizations for (weakly) (x-a)(x-b)-precious rings in terms of (weakly) precious rings. In particular, we prove that if 2 is a unit, then a ring is precious if and only it is weakly precious. Finally, for n∈ℕ, we study (weakly) (xⁿ-x)-precious rings and clarify some of their properties.


Introduction
Throughout this paper all rings are considered associative with an identity. We use U (R), N (R) and Id(R) to denote the the group of units, the set of nilpotent elements and the set of idempotents in R respectively. In 1977 Nicholson, [19], introduced the concept of clean rings as rings in which every element is a sum of a unit and an idempotent. He showed that clean rings must be exchange and they contain unit-regular and semiperfect rings. In [1], Ahn and Anderson defined a ring R to be weakly clean if each element r ∈ R can be written as r = u + e or r = u − e for u ∈ U (R) and e ∈ Id(R). As a variant of (weakly) clean rings, A ring R is said to be (weakly) nil-clean if for every element r ∈ R, there are b ∈ N (R) and e ∈ Id(R) such that (r = b + e or r = b − e) r = b + e . Nil clean rings are in fact a stronger concept than clean rings and it was defined as early as 1988 by Hirano and others, [16]. Then (weakly) clean and nil clean rings were studied extensively by many authors, see for example [15,20,2,13,7,9,12].
For a ring R, we let C(R) denotes the center of R and g(x) be a polynomial in C(R) [x]. Camillo and Simon, [11], defined a ring R to be g(x)-clean if for every r ∈ R , r = s + u where u ∈ U (R) and g(s) = 0. Clearly, clean rings are precisely 246 H. A. Khashan the (x 2 − x)-clean rings. Many properties of g(x)-clean rings have been studied by Fan and Yang, [14]. For a, b ∈ C(R) with b − a ∈ U (R), they characterized clean rings as (x − a)(x − b)-clean rings. Weakly g(x)-clean rings were defined and studied by Ashrafi and Ahmadi, [4] in an analogous way to weakly clean rings. As a special class of g(x)-clean rings, Khashan and Handam, [17], defined and studied g(x)-nil clean rings as rings in which every element is a sum of a nilpotent and a zero of g(x). More recently, Ashrafi, Shebani and Chen, [5], have defined an element a in a ring R to be precious (weakly precious) if there exist an idempotent e, a unit u and a nilpotent b such that a = e + u + b (a = e + u + b or a = −e + u + b ). A ring R is called (weakly) precious if every element in R is (weakly) precious. It has been proved that (weakly) precious rings are proper generalizations of both (weakly) clean and (weakly) nil clean rings. Moreover, it is clear that (weakly) clean rings and (weakly) precious rings coincide if the rings are reduced or commutative. Prompted by this definition, we define and study (weakly) g(x)-precious rings as a generalization of (weakly) precious, (weakly) g(x)-clean and g(x)-nil clean rings. R is called g(x)-precious if there exist a zero s of g(x), u ∈ U (R) and b ∈ N (R) such that a = s + u + b (equivalently, a = −s + u + b). As a general concept, we define a ring R to be weakly g(x)-precious if there are a zero s of g(x), u ∈ U (R) and b ∈ N (R) such that a = s + u + b or a = −s + u + b (in short, we write a = ±s + u + b).
In section 2, we first determine some conditions (weaker than commutativity) under which (weakly) g(x)-clean rings and (weakly) g(x)-precious rings are the same. Then we investigate the structure theorems of (weakly) g(x)-precious rings similar to those of (weakly) g(x)-clean and g(x)-nil clean rings. We deduce the (weakly) g(x)-preciousness of quotient rings, direct product of rings, triangular matrix rings and full matrix rings. Moreover, we characterize (weakly) g(x)-precious rings by modules.
In section 3, we study (weakly) g(x)-precious rings for some special kinds of polynomials g(x) ∈ C(R) [x]. We study (x − a)(x − b)-precious and weakly x(x − b)precious rings where a, b ∈ C(R) and characterize them in terms of precious and weakly precious rings. Among other results, we prove that if 2 ∈ U (R), then the statements R is precious and R is weakly (x 2 − 1)-precious are equivalent. Finally, for n ∈ N, we study some particular properties of (weakly) (x n − x)-precious rings.

Basic Properties of (Weakly) g(x)-Precious Rings
The purpose of this section is to give some examples and investigate some basic properties of (weakly) g(x)-precious rings.
It is clear that any precious ring is weakly precious. The converse is not true since for example simple computations show that both the rings Z 3 and T 2 (Z 3 ) are weakly (x 3 − 1)-precious which are not (x 3 − 1)-precious. While obviously, (weakly) precious rings are (weakly) (x 2 −x)-precious, there are (weakly) g(x)-precious rings that are not (weakly) precious. For example, consider the group ring Z (7) C 3 where C 3 is the cyclic group of order 3 and Z (7) = m k : m,k ∈ Z and gcd(7, k) = 1 . We will see in the next section that Z (7) C 3 is (x 4 − x)-precious which is not precious. It is also clear that every g(x)-clean element in a ring is g(x)-precious. However, the converse is not true in general. For example, it was shown in Theorem (3) in [3] that For a non trivial example, one can easily verify that the ring Z 6 is (weakly) (x 2 − 2x)-precious which is not (weakly) (x 2 − 2x)-clean. Since g(x)-nil clean rings are g(x)-clean by Proposition (2.3) in [17], then every g(x)-nil clean ring is g(x)-precious. The converse is also not true since for example we can verify that the ring Z 10 is an ( For any commutative ring, it is well known that the sum of a nilpotent and a unit is a unit. Therefore, the class of commutative (weakly) g(x)-precious rings is precisely the class of commutative (weakly) g(x)-clean rings.
An element a in a ring R is called strongly nilpotent if for every sequence a = a 0 , a 1 , ..., a i , ... such that a i+1 ∈ a i Ra i , there exists an integer n with a n = 0. It is well known that the prime radical P (R) of R is exactly the set of all strongly nilpotent elements of R. A ring R is called 2-primal if every nilpotent element of R is strongly nilpotent, see [18]. A ring R is called a left (right) quasi-duo ring if every maximal left (right) ideal of R is an ideal. In the following proposition, we clarify other conditions under which g(x)-clean rings and g(x)-precious rings coincide.
Proposition 2.2. Let R be a 2-primal ring or a left (right) quasi-duo ring and g(x) ∈ C(R) [x]. Then R is (weakly) g(x)-clean if and only if R is (weakly) g(x)precious.
Proof. ⇒) : Clear ⇐) : Suppose R is a 2-primal g(x)-precious ring and let a ∈ R. Choose a zero s A ring R is called nil-semicommutative if for any a, b ∈ N (R), ab = 0 implies that aRb = 0. By Lemma (2.7) in [18], any nil-semicommutative ring is 2-primal. Therefore, the equivalence in Proposition 2.2 holds also if R is nil-semicommutative.
For any two rings R and S, we let ϕ : C(R) → C(S) be a ring homomorphism Proposition 2.3. Let R and S be two rings and ϕ : R → S be an epimorphism.
and let t ∈ S. Choose an element a ∈ R such that ϕ(a) = t. Since R is g(x)precious, then a = s + u + b for some zero s of g(x), u ∈ U (R) and b ∈ N (R).
The converse of Corollary 2.4 is true in the case of finite direct product.
On the other side, we next prove the following proposition in the case of weakly g(x)-precious rings.
. Then R is weakly g(x)-precious if and only if each R α is weakly g(x)-precious and at most one R α is not g(x)-precious.
Proof. ⇒) : Suppose R is weakly g(x)-precious. Then for α ∈ Λ, R α is weakly g(x)precious as it is a homomorphic image of R. Suppose for some α 1 and α 2 , α 1 = α 2 , R α 1 and R α 2 are not g(x)-precious. Choose r 1 ∈ R α 1 not of the form ⇐) : Conversely, we assume that R β is a weakly g(x)-precious that is not g(x)precious for a fixed index β ∈ Λ and R α is g(x)-precious for all α = β. Let r = (r α ) ∈ R. Then there exist in R β a unit u β , a nilpotent b β and a zero s β of is again a weakly g(x)-precious decomposition of r. Therefore, r is weakly g(x)precious in R as required. ✷ Proposition 2.7. Let R be a ring, I be an ideal of R and consider the canonical homomorphism ϕ : (2) If R/I is (weakly) g ϕ (x)-precious where I is nil and roots of g ϕ (x) left modulo I, then R is (weakly) g(x)-precious.

Proof. (1) It is clear by Proposition 2.3.
(2) suppose R = R/I is g ϕ (x)-precious and let a ∈ R. Then a = a+I = s+u+b where g ϕ (s) = 0, u ∈ U (R) and b ∈ N (R). Since I is a nil ideal of R, we can easily check that u ∈ U (R) and b ∈ N (R). As the root s of g ϕ (x) lifts modulo I, we may assume also that s ∈ R with g(s) = 0. Hence, a = s + u + b + i for some i ∈ I. If we choose an integer k ≥ 1 such b k = 0, then we get (b + i) k ∈ I. Therefore, b + i ∈ N (R) and so R is g(x)-precious. If R/I is weakly g ϕ (x)-precious, then R is weakly g(x)-precious by a similar approach.
is g(x)-precious. The case of weakly g(x)-preciousness is similar ✷ Since for a commutative clean ring R, the ring of polynomials R[t] is not clean, [15], then R[t] is not precious (and so is not an (x 2 − x)-precious).
Let R be a ring and n ∈ N. Then the matrix ring M n (R) is a C(R)-algebra. Indeed, we can define π : C(R) → M n (R) by π(a) = aI n where I n is the identity matrix. In the next two propositions, we determine when the upper triangular matrix ring T n (R) and the full matrix ring M n (R) are (weakly) g(x)-precious for g(x) ∈ C(R)[x]. Proposition 2.9. Let R be a ring and let g(x) ∈ C(R) [x]. The following are equivalent.
(2) T n (R) is g(x)-precious for all n ∈ N.
Then one can easily see that . Hence, T n (R) is g(x)-precious. . By assumption, we may write and s 1 , s 2 are zeros of g(x). Therefore, R is g(x)-precious. ✷ Proposition 2.10. Let R be a ring, n ∈ N and g(x) ∈ C(R)[x]. If R is g(x)precious, then so is the matrix ring M n (R).
Proof. We use the induction on n. For n = 1, the result is clear. Let n 1 and suppose the statement is true for M n−1 (R). Let A ∈ M n (R) and write A = A * X Y a in block form where a ∈ R and A * ∈ M n−1 (R). Since M n−1 (R) is g(x)-precious, then we can find U * ∈ U (M n−1 (R)), B * ∈ N (M n−1 (R)) and a zero S * of g(x) such that then U is a unit in M n (R). Since also clearly B ∈ N (M n (R)) and g(S) = 0, then M n (R) is g(x)-precious.
. Clearly s 0 0 s is a zero of g(x) and u 0 0 u ∈ U (R(M )). Moreover, simple computations show that b m 0 b ∈ N (R(M )). Therefore, R(M ) is weakly g(x)-precious. The g(x)-preciousness equivalent is similar. ✷

Special Types of (Weakly) g(x)-Precious Rings
In this section we study some special types of (weakly) g(x)-precious rings. First, we consider an (x − a)(x − b)-precious (weakly x(x − b)-precious) ring R where a, b ∈ C(R) and give a characterization of such a ring in terms of precious (weakly precious) rings. Other types of polynomials such as x n − x, x n + x and x n − 1 are also considered.
We start by the following main theorem.
Theorem 3.1. Let R be a ring and a, b ∈ C(R).
Proof. (1) ⇒) : Suppose R is precious and let r ∈ R. As b − a ∈ U (R), then we may write r−a b−a = e + u + n where e ∈ Id(R), u ∈ U (R) and n ∈ N (R). Thus, Therefore, R is precious.
(2) Suppose R is weakly precious with b ∈ U (R). For r ∈ R, we write r b = ±e + u + n where e is an idempotent, u is a unit and n ∈ N (R). Then, r = ±eb + ub + nb where clearly ub ∈ U (R), nb ∈ N (R) and eb is a root of x(x − b). Hence, R is weakly x(x − b)-precious. The other implication is almost similar and left to the reader. ✷ If we take b = −1 in the previous theorem, we conclude that R is (weakly) precious if and only if R is (weakly) x(x + 1)-precious.
We note that the above equivalence of weakly precious rings and weakly (x − a)(x−b)-precious rings is a ring property. This means that this equivalence holds for a ring R but it may fail for a single element. For example, 3 is weakly (x−1)(x−2)precious which is not weakly precious in Z.
We recall that a unit in a ring R is called unipotent if it is of the form 1 + b where b ∈ N (R). For any ring R and n ∈ N, the set of all elements of R that can be written as a sum of no more than n units is denoted by U n (R). If R = U n (R), then R is called an n-good ring. Such rings were studied extensively by many authors, see for example [21] and [22]. Next, we generalize this concept as follows.
Definition 3.2. Let R be a ring. Then R is said to be involution 3-good if for every r ∈ R, we have r = u + w + v where u ∈ U (R), w ∈ 1 + N (R) and v 2 = 1.
Proof. Suppose R is involution 3-good and let k ∈ N. For r ∈ R, write r − k = u + w + v where u ∈ U (R), w ∈ 1 + N (R) and v 2 = 1. Then, In the following Theorem, we characterize involution 3-good rings in term of precious and g(x)-precious rings.
Theorem 3.4. Let R be a ring for which 2 ∈ U (R). The following are equivalent: (1) R is precious.
The equivalence of statements (1), (2) and (3) in Theorem 3.4 can be generalized if n ∈ U (R) for any n ∈ N. The proof is similar.
Theorem 3.5. Let R be a ring and n ∈ N for which n ∈ U (R). The following are equivalent: (1) R is (weakly) precious.
Example 3.6. For any Continuous or discrete R-module M where n ∈ U (R), the endomorphism End R (M ) is clean, [10], (and so precious).
Example 3.7. For any strongly zero dimensional topological space X, the ring of all continuous real valued functions C(X) is clean with n ∈ U (C(X)), [6]. So, C(X) and M k (C(X)) are (x 2 − nx)-precious for any k ∈ N by Theorem 3.5 and proposition 2.10.
Example 3.8. let n, k ∈ N, F be a field with character char F = c ∤ n and V be an infinite dimensional vector space over F . Let R be a subring of End F (V ) generated by the identity and the finite rank transformations. Then R is clean with n ∈ U (R), [15], and so R and M k (R) are (x 2 − nx)-precious.
In the remaining results of this section, we will concern about (x n − x)-precious rings where n ∈ N. An element p in a ring R is called potent if p n = p for some n ∈ N. It is clear that any idempotent element in R is potent and so any precious ring is (x n − x)-precious. However, the converse is not true in general. For example consider the group ring Z (7) C 3 where C 3 is the cyclic group of order 3 and Z (7) = m k : m,k ∈ Z and gcd(7, k) = 1 . Then by Theorem (3.1) in [25], Z (7) C 3 is (x 4 − x)-clean (and so (x 4 − x)-precious). On the other hand Z (7) C 3 is not clean by Example (1) in [15]. Since also Z (7) C 3 is reduced, then Z (7) C 3 is not precious. A ring R is said to be semipotent if each left ideal of R that is not contained in the Jacobson radical contains a non zero idempotent. In [23], Wood proved also that Z (7) C 3 is not semipotent and so an (x 4 − x)-precious need not be semipotent.
Recall that for a ring R and n ∈ N, R is called n-clean ring if for any a ∈ R, there is e ∈ Id(R) and u 1 , u 2 , ..., u n ∈ U (R) such that a = e + u 1 + u 2 + ... + u n . Proposition 3.9. Let R be a ring and let n ∈ N. If R is (x n − x)-precious, then R is 3-clean. If moreover, 2 ∈ U (R), then R is 4-good.
Proof. Suppose R is (x n − x)-precious and let a ∈ R. Then a − 1 = s + u + b where u ∈ U (R), b ∈ N (R) and s = s n is a potent element of R. Since it is well known that every potent element is clean, then s = e + v where e ∈ Id(R) and v ∈ U (R). Therefore, a = e + v + u + (b + 1) is a 3-clean element in R and R is 3-clean. Now, suppose 2 ∈ U (R) and write a+1 2 − 1 = s + u + b where u ∈ U (R), b ∈ N (R) and s = s n , then similarly we can write a = (2e − 1) + 2v + 2u + 2(b + 1) where u, v ∈ U (R), b ∈ N (R) and e ∈ Id(R). It is clear that the four terms in this expansion of a are units and so R is 4-good. ✷ Proposition 3.10. Let R be a ring, n ∈ N and g(x) = x n + a n−1 x n−1 + ...
In the following proposition, we can see that (weakly) (x n − x)-precious rings are the same as (weakly) (x n + x)-precious rings for any even positive integer n. Thus, we conclude that the ring Z (7) C 3 is also (x 4 + x)-precious.
Proposition 3.11. Let R be a ring, k ∈ N and a, b ∈ R. Then R is (weakly) (ax 2k + bx)-precious if and only if R is (weakly) (ax 2k − bx)-precious.
⇐) : Conversely, Suppose R is (ax 2k − bx)-precious and let r ∈ R. Write −r = s+u+n where u ∈ U (R), n ∈ N (R) and as 2k −bs = 0. Then r = (−s)+(−u)+(−n) where −u ∈ U (R), −n ∈ N (R) and a(−s) 2k + b(−s) = as 2k − bs = 0. Therefore, R is (ax 2n + bx)-precious. The weakly case is almost the similar. ✷ The equivalence in proposition 3.11 need not be true for odd powers. For example it is easy to check that the field Z 3 is an (x 3 − x)-precious which is not (x 3 + x)-precious. Theorem 3.12. Let R be a (weakly) (x n − x)-precious ring, a ∈ R and n ≥ 2. Then either a is a (weakly) (x n−1 − 1)-precious element or there is b ∈ N (R) such that (a − b)R and R(a − b) contain non zero idempotents.