Graded fuzzy ring and control picture by coloring fuzzy location

In this article, the definition of graded fuzzy ring by group and some of related theorems has been investigated. Among it is the relation between a graded fuzzy ring with crisp graded ring by level sets and gain the set of cosets a graded ring on a fuzzy ideal with a special condition is graded ring. After that be graded fuzzy ring on a group ring and its result on Laurent polynomial is showed. Finally we prove a graded fuzzy ring for a specific group ring that is fuzzy on group elements and then we control a system of image processing by setting of standard colors.


Introduction
Lotfi A. Zadeh [13] in 1965 introduced the notion of a fuzzy set.Rosenfield applied this concept in the group theory [10].In commutative algebra W. J. Liu [5] opened the way towards the development of fuzzy algebraic structures by introducing the notions of fuzzy normal subgroup, fuzzy subring and the product of fuzzy sets.Liu [12] introduced the notion of a fuzzy ideal of a ring.N. Kuroki [2] demonstrated the utility of the notion of the fuzzy set in the more general setting of semigroups.The concepts of fuzzy fields and fuzzy linear spaces were introduced by S. Nanda [8].Malik and Mordeson presented direct sum of fuzzy rings and fuzzy ideals [7].Graded algebras are much used in commutative algebra and algebraic geometry, homological algebra and algebraic topology.One example is the close relationship between homogeneous polynomials and projective varieties.Eslami explained a concept of graded fuzzy rings on natural numbers by direct sum of fuzzy subgroup and then found a way for fuzzification polynomial rings [1].This article introduce a graded fuzzy ring by an arbitrary group.We prove that how a fuzzy ring is graded whenever all of its level sets be geaded.We effort open a way toward introduction a fuzzy group ring by graded fuzzy ring and induce the results to Laurent polynomial that we extend Eslami work.We show a group ring with fuzzy elements of group can be a graded fuzzy ring.Finally, by definition of binary operation on standard colors system, we display a pixel color by a fuzzy element of group ring and control the resolution of pixel with a control function.Afterwards the picture will be prossece by a graded fuzzy ring.

Priliminaries
In the all parts of this article R is a commutative ring and unital.Let µ be a fuzzy subset in R that µ(0) = 1.If for every x, y ∈ R, we have µ(x − y) ≥ min{µ(x), µ(y)}, then µ is a fuzzy subgroup of R and we call fuzzy subgroup µ a fuzzy ring (fuzzy ideal) of R if µ(xy) ≥ min{µ(x), µ(y)} (µ(xy) ≥ max{µ(x), µ(y)}).Suppose that I is a nonempty set.For x, x i ∈ R with i ∈ I, we mean from x = ∑ i∈I x i that x can be written as element summation of x i which x i s are zero except finite numbers.Definition 2.1.[7] Let {µ i | i ∈ I} be a family of fuzzy subsets on R. Fuzzy subset ∑ i∈I µ i on R is defined as below, ( Remark 2.1.Let {µ i | i ∈ I} be a family of fuzzy subgroup, fuzzy ring or fuzzy ideal on R, then easily it can be shown that ∑ i∈I µ i is fuzzy group, fuzzy ring or fuzzy ideal on R, respectively.If for every x ∈ R we set x j = x if i = j and x j = 0 if j ̸ = i, then by the definition of ∑ j∈I µ j , we have, Definition 2.2.[1] Let µ i and µ with i ∈ I be fuzzy subsets on R. Then R is called weak direct sum of In this case µ = ⊕ i∈I µ i is written.
Definition 2.3.[7] Let µ be a fuzzy subset of R. We define, the fuzzy subset µ is called a finite value fuzzy subset on R, if Im(µ) is finite set .Proposition 2.1.[7] Let µ be a fuzzy ring (fuzzy ideal), then µ * and µ * are subrings (ideals) of R. Definition 2.4.[7] Let µ and ν be two fuzzy subsets on R. Fuzzy subset µν on R is defined as following, Proposition 2.2.[7] Let µ and ν be fuzzy rings (fuzzy ideals) on R. Then µν is fuzzy ring (fuzzy ideal) on R. Definition 2.5.[4] Let µ be a fuzzy ideal of R and let x ∈ R. Then the fuzzy subset x + µ of R defined by is termed as the fuzzy coset determined by x and µ.The set of all fuzzy cosets of µ in R is a ring under the binary operations and it is denoted by R/µ.We call it the fuzzy quotient ring of R induced by the fuzzy ideal µ.
Definition 2.6.[9] Ring R is called graded by (G, o) if there exists a family of {R g } g∈G from subgroups of R, which R = ⊕ g∈G R g , and for every g, g International Scientific Publications and Consulting Services Definition 2.7.[1] Fuzzy ring µ on R is graded by natural number when fuzzy groups µ i , i = 0, 1, 2, • • • are existed on R such that, • µ i µ j ⊆ µ i+ j for all i, j = 0, 1, 2, • • • .
3 graded fuzzy ring Definition 3.1.Let µ be a fuzzy ring on R and (G, o) be a group (semigroup).We say µ is a graded fuzzy ring by G, if there exists a family {µ g } g∈G from fuzzy subgroups on R that µ = ⊕ g∈G µ g , and, for every g, g ′ ∈ G, we have µ g µ g ′ ⊆ µ gog ′ .Example 3.1.Suppos that R is a graded ring by G on {R g } g∈G from its subgroups.Then we define µ g : R g → [0, 1] by µ g (x) = 1 if x ∈ R g and if not µ g (x) = 0. Obviously, it can be shown that fuzzy ring µ : R → [0, 1] defined by µ(x) = 1 is a graded fuzzy ring by G on family of {µ g } g∈G from fuzzy subgroup on R.
The example above shows that our definition from graded fuzzy ring by group has been extended.Theorem 3.1.Let {µ g | g ∈ G} be a family of fuzzy groups on R and µ be a fuzzy ring on R with µ = ∑ g∈G µ g .Then µ is a graded fuzzy ring by group G if and only if for every α ∈ Proof.Let {µ g } g∈G be a family of fuzzy subgroup on R that µ = ⊕ g∈G µ g and for every g, g contradiction by the definition of graded fuzzy ring.So x = 0 and , and for all 0 ≤ i ≤ n, we have The proof is completed.
We set S = ⊕ g∈G R g µ g * .Then we claim that S become a ring with the following product operation.
For all ).
Since R is a graded ring, if for x 1g , x 2g ∈ R g and y 1g ′ , y 2g ′ ∈ R g ′ , we have . Hence our claim is correct.On the other hand, S is a graded fuzzy ring by group G, beacause . Therefore the proof will be completed.

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From the induction of gradation µ by G, we result µ h µ h ′ ⊆ µ hoh ′ , for every h, h ′ ∈ H. Now, for all x ∈ R we have, 4 Examples of graded fuzzy ring and applications Example 4.1.Suppose µ be a fuzzy ring on R. For every group G, we set µ e = µ and µ g = 1 {0} for e ̸ = g ∈ G. Then µ is a graded fuzzy ring called trivial gradation.
We remind if G is a group and R is a ring, then we present the set of maps f : G → R with finite support as ℑ.Scalar product α f is defined as x → α • f (x), for α ∈ R, and also for every f , g ∈ ℑ, we introduce, It is shown that (ℑ, +, •) forms a ring.Now, let µ be a fuuzy ring on R. Then we define ν and ν x fuzzy subsets on ℑ for x ∈ G as Remark 4.1.It most be paid attention that ν( f ) and ν x ( f ) are well defined, because f ∈ ℑ is a map with finite support.
Proposition 4.1.The fuzzy set ν is a fuzzy ring on ℑ and ν x is a fuzzy subgroup, for all x ∈ G .
Proof.For all f , g ∈ ℑ, since f , g are maps with finite support that obtain, Hence ν is a fuzzy ring on ℑ.Now for every x ∈ G and f , g ∈ ℑ, we have If f (y) = 0 and g(y) = 0, then f (y) = g(y).Therefore we result If the revised definition, then we obtain that ν = ⊕ x∈G ν x and ν x ν y ⊆ ν xy for all x, y ∈ G.
Example 4.2.Let R[G] be a group ring relate to R as ring and G as group and µ a fuzzy subgroup on R. Then we define According to the previous description, we can easily obtain that R[G] ≃ ℑ.So ν, the fuzzy subset defined on R[G] is a graded fuzzy ring by G, Which we call a fuzzy group ring.

Example 4.4. For every H subgroup of G, fuzzy group ring on R[H] induced of ν, fuzzy group ring on R[G] is
For a ring R and a group G, a fuzzy group ring on elements of G over R is notation by R[G [0,1] ] and defined as follows, , n ∈ N

}
For making a ring a criterion for the assesment of equality between two elements most be presented.In other word we most state the summation and production on the elements of R[G [0,1] ], in such an extend that the necessary condition in the definition of the ring be provided.Therefore we consider, if and only if n = m and After a suitable index, The summation act on R[G [0,1] ] introduced as below, if set International Scientific Publications and Consulting Services and then we define, the multiplication "o" on elements of R[G [0,1] ] according to the following method described, for every g, h ∈ G and α, β ∈ [0, 1], we define i.
Example 4.5.suppose µ be a fuzzy ring on R. Then we introduced ν, the fuzzy set on R[G [0,1] ] as below, for every l i ≥ 1, we define, Similarly to what was done before, ν is a fuzzy ring on R[G [0,1] ] and ν and we can prove that, shows the resolution level of a picture with variety of colors on colorful pixel seperation.Now the situation control function of picture resolution as µ : f [S [0,1] ] → [0, 1] can be set by purpose processing.For example if purpose processing be bolding picture resolution based on reddish colors, we can use the control function as µ α : f [S [0,1] ] → [0, 1] with The act of this control function is vanishing the resolution of reddish colors with less tendentions of α and in this way the red shadow in a pixel will be omitted and the red color of picture will bold.
For the whole of picture, the control function is as following, )).

conclusion
The set S [0,1] make it possible for us to find the location of different places of a picture based on the amount and composition of the used color and applied control function by this location.
As it has seen function µ is an applied example on fuzzy situation set S [0,1] .The last function is an description of any situation, property or description of spectrum based on an essential system.The description of particles around the atom by the properties of the core like mass, volume, tempreture and etc, or the situation of not clear places in a satelite image from a geographical place based on the quality of the nearest clear place in the image, are the options that can be processed with the function µ.Actually function µ is the processing function related to marginal areas of a phenomenon.
Theorem 3.2.Let R be a graded ring by G with family of ideals {R g } g∈G and µ be a fuzzy ideal on R. If 1 > If R be an Artinian ring.Then every fuzzy ideal on R is finite value by [[6] , theorem 3.2] and corollary 3.2.Proposition 3.1.Let µ = ⊕ g∈G µ g be a graded fuzzy ring by group G and H be an arbitrary subgroup of G. Then µ H = ⊕ h∈H µ h is a grade fuzzy ring by group H and µ H ⊆ µ.
Corollary 3.2.Let R be a graded ring with family of ideals.Then for every finite value fuzzy ideal µ, we gain R µ is graded ring.Proof.It is clear by theorem 3.2.Corollary 3.3.Let R be a graded Artinian ring with family of ideals.Then R µ is graded ring for every fuzzy ideal µ on R. Proof.Proof.As regards remark 2.1, µ H is a fuzzy subgroup on R. It is sufficient to show that µ H (xy) ≥ min{µ H (x), µ H (y)} for all x, y ∈ R.

Table 1 :
We consider S as standard colors system which is produced by three basic standard colors Green, Blue and Red.Binary operation of standard colors is at the table.1.Table of combination act on standard color RGB In the table.1,RG is the result of combination of Red and Green.Since in the mixture of the two colors the priority and order is not important , the operation above is commutative.Obviously it can be shown that (S, o) is a semigroup.Now forγ i ∈ [0, 1] that 1 ≤ i ≤ 3, we mean R γ 1 G γ 2 B γ 3as a color which is a member of standard color Red γ 1 , standard color Green γ 2 and standard color Blue γ 3 .In this case we consider standard color Red is R 1 , standard color Green G 1 and standard color Blue B 1 .As regards defined multiplication on S [0,1] = {R γ 1 G γ 2 B γ 3 |γ 1 , γ 2 , γ 3 ∈ [0, 1]} in the last subject, which expressed the operation of "o" on S in table above, is well defined.Suppose f : S [0,1] → R expresses the level of different color spectrum resolution.So, . International Scientific Publications and Consulting ServicesExample 4.6.