Neutrosophic Quotient Algebra

The algebraic properties of neutrosphic ideals over algebra, isomorphism properties of neutrosophic ideal and neutrosophic modules over algebra are discussed in this paper. Some of the charactrisations of Neutrosophic quotient algebra are derived and the role of algebraic structures is studied in the context of neutrosophic set. This paper expands the definition of quotient algebra within the context of neutrosophical set


1.Introduction
In classical set theory, the membership of elements in a set is assessed in binary terms 0 and 1; according to a bivalent condition-an element either belongs or does not belong to the set. As an extension, fuzzy set theory permits the gradual assessment of the membership of elements in a set. A fuzzy set A in X is characterised by a membership function which is associated with each element in X, a real number in the interval [0,1]. Lotfi A Zadeh [1] introduced a theory whose objects fuzzy sets-are sets with imprecise boundaries which allow us to represent vague concepts and contexts in natural language. Fuzzy set theory is limited to modelling a situation involving uncertainty. As an extension of fuzzy set concept, the theory of intuitionistic fuzzy sets introduced whose elements have degree of membership and non membership. Intuitionistic fuzzy sets have been introduced by Krassimir Atanassov [2] as an extension of Lotfi Zadeh's notion of fuzzy set. Let us have a fixed universe X and A is a subset of X .The intuitionistic fuzzy set can be defined as .  for membership and for non membership, which belongs to the real unit interval [0,1] and sum belongs to the same interval.
Neutrosophy is a new branch of philosophy and logic introduced by Florentin Smarandache [3,4] in 1995 which studies the origin and features of neutralities in nature. Each proposition in Neutrosophic logic is approximated to have the percentage of truth (T), the percentage of indeterminacy (I) and the percentage of falsity (F). So this Neutrsophic logic is called generalization of classical logic, conventional fuzzy logic, intuitionistic fuzzy logic and interval valued fuzzy logic. This mathematical tool is used to handle problems like imprecise, indeterminate and inconsistent data. The use of neutrosophic theory becomes inevitable when a situation involving indeterminacy is to be modelled. The introduction of Neutrosophic theory has led to the establishment of the concept of neutrosophic algebraic structures in this article [5,6]. For more information on real applications of neutrosophic theory, the readers can see [13][14][15] The main objective of the neutrosophic set is to narrow the gap between the vague, ambiguous and imprecise realworld situations. Among the different branches of applied and pure mathematics, abstract algebra was one of the first few area where research was conducted using the concept of neutrosophic set. Initially, B. Vasantha Kandasamy and Florentin Smarandache [7] introduced and applied fundamental algebraic neutrosophic structures. This paper focuses on algebra over a field , quotient algebra over a field and algebraic structures ideal in neutrosophic domain and derive some algebraic properties. This paper focuses on algebra over a field , quotient algebra over a field and algebraic structure ideal in neutrosophic domain and derives some algebraic properties.

Preliminaries
Abraham Robinson [8] introduced the non-standard analysis in the 1960s, a formalization of the analysis and a branch of mathematical logic that describes the infinitesimals. Informally, an infinitesimal is an infinitely small number. , where 0 is its standard part and its non-standard part. Then, we call] -0, 1+ [a non-standard unit interval. Obviously, 0 and 1, and analogously non-standard numbers infinitely small but less than 0 or infinitely small but greater than 1, belong to the non-standard unit interval. Generally the left and right borders of a, are vague, imprecise and themselves being a non standard subsets.
Definition 2.1 [9,10] A Neutrosophic set A on the universal set X is defined as t i f are known as Neutrosophic components which are subsets of . A Neutrosophic set A can be written as

Since the membership function
defined in real life and scientific applications are from X in to the unit , a Neutrosophic set A will be denoted by a mapping defined by , xis hard work, y is capability and z is knowledge in particular area. They are obtained from the questionnaire of some domain experts about the question good researcher. A is single valued Neutrosophic set of X defined by [3,9] Let A and B be two Neutrosophic sets on X. Then ii) The union of A and B is denoted by C A B   and defined as iii) The intersection of A and B is denoted by C A B   and defined as [11] An algebra is an algebraic structure which consist of a set, together with multiplication, addition and scalar multiplication by elements of underline field and satisfies the axioms implied by vector field and bilinear. An algebra over a field is a vector space equipped with bilinear product. Definition 2.4 [12] Let V be a vector space over a field F equipped with binary operation from Then V is an algebra over a field F if the following conditions hold

Neutrosophic quotient algebra
This section defines neutrosophic quotient algebra over a field and derive some elementary properties by extending the concept of algbra over a field in neutrosophic set.    (   2  1  2  1  2  1  2  1  2  1  2  1   a  f  a  f  a  a  f  a  i  a  i  a  a  i  a  t  a  t  a  a N -ideal and Y be an algebra over a field X, then Y y  , define neutrosophic Similarly we can prove that A y A y    1 2 ... (4) From (3) and (4) ) 0 ( ) (

Conclusions
Neutrosophic quotient algebra is one of the generalizations of quotient algebra. This paper has developed a combination of an algebraic structure , quotient algebra with neutrosophic set theory. Neutrosophic quotient algebra becomes a key element in the study of neutrosophic quotient modules of an R-module and their properties. This study leads to algebraic nature of neutrosophic algebraic structure and the evolution of new neutrosophic algebraic structures.
Funding: "This research received no external funding" Conflicts of Interest: "The authors declare no conflict of interest."