Soft normed rings

Molodtsov introduced the concept of soft sets, which can be seen as a new mathematical tool for dealing with uncertainty. In this paper, we initiate the study of soft normed rings by soft set theory. The notions of soft normed rings, soft normed ideals, soft complete normed rings are introduced and also several related properties and examples are given.


Preliminaries
In this section we recall some basic notations in soft set theory. Let U be an initial universe set and E be a set of all possible parameters under consideration with respect to U. The power set of U is denoted by P(U). Molodstov defined the notation of a soft set in the following way: Definition 1 (Molodtsov 1999) A pair (F, A) is called a soft set over U, where F is a mapping given by F : A → P(U).
Definition 2 (Maji et al. 2003) For two soft sets (F, A) and (G, B) over a common universe U, we say that (F, A) is a soft subset of (G, B) if 1. A ⊆ B and 2. G(ε) ⊆ F (ε), ∀ε ∈ B.
We write (F , A) ⊆ (G, B). In this case (G, B) is called a soft superset of (F, A).
Definition 3 (Maji et al. 2003) Let (F, A) and (G, B) be two soft sets over a common universe U. The union of (F, A) and (G, B) is defined as the soft set (H, C) satisfying the following conditions: This is denoted by (F , A)∪ (G, B) = (H, C).
Definition 4 (Maji et al. 2003) Let (F, A) and (G, B) be two soft sets over a common universe U. The intersection of (F, A) and (G, B) is defined as the soft set (H, C) satisfying the following conditions: Definition 5 (Das and Samanta 2012) Let R be the set of real numbers and B(R) be the collection of all non-empty bounded subsets of R and let A be taken as a set of parameters. Then a mapping F : A → B(R) is called a soft real set. If a soft real set is a singleton soft set, it will be called a soft real number and denoted by r,s,t etc. 0,1 are the soft real numbers where 0 (e) = 0,1(e) = 1 ∀e ∈ A, respectively.

Soft normed rings
Definition 8 Let H be an associative soft ring with 1 . A soft norm H is a function �.� : H → R(A) that satisfies the following conditions for all x,ỹ ∈ H.
The soft ring H with a soft norm ‖.‖ on H is said to be a soft normed ring and is denoted by (H, . , A) or (H, . ).
Assume now that the soft operators T nx converge to Tx in the soft norm of the space H for every x ∈ H. By the soft continuity of multiplication with respect to the first factor, we then have:

Conclusion
Normed rings have previously been described in the classical sense. In this study, normed rings are defined on soft sets for the first time. This may lead to an ample scope on soft normed rings in the soft set setting. In this paper, we defined a soft normed ring. We then investigated some related properties and some theorems. To extend this work, one can study the properties of soft normed rings in other algebraic structures and fields.
Authors' contributions VU, MS and NO completed the main part of this article. All authors read and approved the final manuscript.