Normed quotient rings

In this article, we introduce the notions of normed quotient ri g, normed quotient subring, normed quotient ring homomorphism, normed quotient ring natural homomorphism a nd investigate some of their related properties.


Introduction
Ziembowski [27] exists a duo ring R such that its classic al right ring of quotients Q r Cl (R) is left duo and not right duo. Building on this construction, we will construct a duo ring with classical right ring of quotients which is neither right nor left duo. For group rings the classical quotient ring has been studied by Herstein and Small [13], Passman [18,19],Smith [21], and Smith [22], and the maximal quotient ring has been studied by Burgess [7]. Formanek [9] investigates the relationship of the maximal quotient rings of group rings, subgroup rings, and the centers of group rings. The object is to obtain for the maximal quotient ring analogues of theorems of Passman and M. Smith on the classical quotient ring. Ara [3] defined by Z(R) the center of R and by Q(R) its maximal right quotient ring. Utumi [26] R has always a unique maximal left quotient ring, and moreover the maximal left quotient ring of a total matrix ring of finite degree over R is a total matrix ring of the same degree over the maximal left quotient ring of R. Johnson [16] it is assumed that each ring has a zero right singular ideal and given the quotient structure of a ring having a vanishing right and left singular ideal. Gary et al. [6].
For an arbitrary ring R we completely characterize when Q(R), the maximal right ring of quotients of R, is a direct product of indecomposable rings and when Q(R) is a direct product of prime rings in terms of conditions on ideals of R. They also investigated the connections between the ideal structure of an arbitrary ring R and the ideal structure of Q(R). Francis [25] A ring F has a semi simple maximal right quotient ring Q if and only if Z(R R ) = 0 and dimR R is finite, where a right F−module M is of finite dimension if every direct sum of submodules of M has only finitely many nonzero summands. Bavula [4] proved existence of the largest left quotient ring Q 1 (R), i.e. Q 1 (R) = S 0 (R) (−1) R where S 0 (R) is the largest left regular denominator set of R. Bavula [5] defined the largest strong left denominator set T l (R) of R, the largest strong left quotient ring Q s l (R) := T l (R) (−1) R of R and the strong left localization radical l s R of R, and to study their properties. Shilov [23] defined on decomposition of a commutative normed ring in direct sums of ideals. Gelfand et al. [10] defined commutative normed rings. Freundlich [8] introduced completely continuous elements of a normed ring.
Raikov [20] defined to the theory of normed rings with involution. Naimark et al. [17] defined normed rings. Shilov [24] defined analytic functions in a normed ring. Jarden [14] defined normed rings in 2011 and is studied norms . of c 2018 BISKA Bilisim Technology associative rings are generalizations of absolute values |.| of integral domains. Jung et al. [15] introduced quasi-commutative as a generalization of commutative rings. They also provide several sorts of examples by showing the relations between quasi-commutative rings and other ring properties which have roles in ring theory. Alvir [1] explored the compressed zero-divisor graph associated with quotient rings of unique factorization domains. Alizadeh [2] for n ≥ 3, they showed that every local derivation from M n (R) into M n (M) is a derivation. Genc [11] given a construction of quotient gamma rings of prime gamma rings in the sense of Nobusawa and to study the some properties of quotient gamma rings of the prime gamma rings. In this paper, our work is organized as follows. We define normed quotient ring using normed spaces. In the section 2, well known results of some preliminaries are given. In section 3, we defined normed quotient ring, normed ideal, normed prime ideal, normed homomorphism, normed natural homomorphism and investigate some of their related properties.

Preliminaries
Definition 1. [17]. The norm . , which is defined on rings, is a generalization of the absolute value |.| defined on integral domains. On rings, the standard rule |x.y| ≤ |x|.|y| is replaced by x.y ≤ x . y . Definition 2. [14]. Let A be commutative ring with 1. An ultrametric absolute value of A is a function || : A → R satisfying the following conditions:  If f is not onto, then A/I is isomorphic to a subring of S/ f (I).
It is easy to show that the norm conditions.
Therefore, the quotient rings satisfy the norm conditions. Proof. Let x, y ∈ A,

Theorem 3. Let A be the normed ring and I be an ideal. Then the mapping . A/I : A/I → S x + I =x ∈ A/I ,defined by
is an onto homomorphism. . A/I is called the normed quotient ring homomorphism.
be a commutative diagram. It is enough to define the normed homomorphism f ′ : Thus f ′ = g and f ′ is unique.

Theorem 5. Let A be a normed ring and I be an ideal of A. Let
is an onto homomorphism whose kernel is I. This normed homomorphism is called the normed natural homomorphism from A onto A/I.
This shows that α is a normed homomorphism. Also bȳ we have Ker(α) = I.

Example 2.
Let A be a normed ring and I be an ideal of A. Show that A/I is a normed commutative quotient ring if and only if for all x, y ∈ A, x.y − y.x ∈ I. Solution:

Theorem 6. Let f : A → S be an onto normed ring homomorphism and I be a (normed) ideal of A . If Ker f ⊆ I, then
A/I ∼ = S/ f (I).

If f is not onto, then A/I is isomorphic to a normed subring of S/ f (I) .
Proof. Letφ : S → S/I be the natural normed homomorphism. Take the normed homomorphism g : A → S/ f (I) such that g = φ f . Then ∀r ∈ A, g(r) = φ ( f (r)) = f (r) A/I g is the composition of two onto normed homomorphisms and hence is onto.
Clearly, I ⊆ f −1 ( f (I)). On the other hand as Ker f ⊆ I, This shows that f −1 ( f (I)) ⊆ I . Therefore,Kerg = f −1 ( f (I)) = I. By the Normed Homomorphism Theorem, we have If f is not onto, it is easy to see that A/I is isomorphic to a normed subring of S/ f (I).

Theorem 7. Let A be a commutative normed ring with unity and I ⊆ A be an ideal. I is a normed prime ideal⇔ A/I is an integral domain.
Proof. (⇒) Let I be the prime ideal. As I = A, A/I has at least two elements. Takex,ȳ ∈ A/I and recall that I is the zero of A/I.x .ȳ = I x.y + I = I ⇒ (x + I)(y + I) = I ⇒ x.y ∈ I ⇒ x ∈ I or y ∈ I (since is a prime ideal) ⇒x = I orȳ = I Thus, A/I doesn't have any zero divisors. As A is commutative and has a unity, so does A/I. This concludes that A/I is an integral domain. showing that I is a normed prime ideal.

Conclusion
In this paper, we examine the algebraic properties of normed quotient ring in ring structures. Some related notions, e.g the homomorphism on normed quotient ring, normed quotient subring, normed quotient ideal are proposed here. This concept will bring a new opportunity in research and development of normed ring theory.