In this article, we introduce and discuss the definitions of fuzzy (m, n)-sub near rings, anti-fuzzy (m, n)- sub near rings, fuzzy ideals, and anti-fuzzy ideals in (m, n)-near rings. We establish several fundamental properties and prove key theorems related to these fuzzy structures. Additionally, illustrative examples are provided to enhance understanding and demonstrate the applicability of the concepts.

In this paper,  first  we introduce the notions of k-nilpotent (solvable) ideals  and k-nilpotent BCK-algebras. Specially, we show  that every commutative ideal is 1-nilpotent (solvable). Second, we state  an equivalent condition to k-nilpotency (solvablity) ideals and  BCK-algebra. Finally,  we study n-fold 2-nilpotent (solvable) ideals and BCK-algebras as a generalization of n-fold commutative ideals and BCK-algebras, and we study the relation between these two concepts. Basically, we compare 2-nilpotent and solvable ideals (BCK-algebras).

Let $R$ be a commutative ring with non-zero identity, and $\delta :\mathcal{I(R)}\rightarrow\mathcal{I(R)}$ be an ideal expansion where $\mathcal{I(R)}$ is the set of all ideals of $R$. In this paper, we introduce the concept of $\delta$-$n$-ideals which is an extension of $n$-ideals in commutative rings. We call a proper ideal $I$ of $R$ a $\delta$-$n$-ideal ifwhenever $a,b\in R$ with$\ ab\in I$ and $a\notin\sqrt{0}$, then $b\in \delta(I)$. For example, an ideal expansion $\delta_{1}$ is defined by $\delta_{1}(I)=\sqrt{I}.$ In this case, a $\delta_{1}$-$n$-ideal $I$ is said to be a quasi $n$-ideal or equivalently, $I$ is quasi $n$-ideal if $\sqrt{I}$ is an $n$-ideal. A number of characterizations and results with manysupporting examples concerning this new class of ideals are given. In particular, we present some results regarding quasi $n$-ideals. Furthermore, we use $\delta$-$n$-ideals to characterize fields and UN-rings.

Let f:A→B be a ring homomorphism and let Jbe an ideal of B. We proved some results concerning n-ideals and(2; n)-ideals of A⋈^f J. Then we recall a proper ideal I of A as √(δ(0))-ideal if ab ϵ I then b ∈ I or a ∈ √(δ(0)) for every a; b ∈ A. We investigate some properties of √(δ(0))-ideal with similar n-idealsand J-ideals.

Let R be a commutative ring with identity. A proper submod-ule N of an R-module M is an n-submodule if rm 2 N (r 2 R; m 2 M) with r =2 p AnnR(M), then m 2 N. A number of results concerning n-submodules are given. For example, we give other characterizations of n-submodules. Also various properties of n-submodules are considered.

Coupled N-structures are introduced, and its application is discussed in BCK/BCI- algebras. The notions of a coupled N-subalgerba, a coupled N-ideal and a coupled NC- ideal are introduced, and their relations are investigated. Characterizations of a coupled N-ideal and a coupled NC-ideal are discussed. Conditions for a coupled N-subalgerba to be a coupled N-ideal are considered.

Let R be a commutative ring with identity. A proper ideal P of R is an (n - 1, n)-Fm-prime ((n - 1, n)-weakly prime) ideal if a1, … , anÎR, a1 … anÎP\P m (a1 … anÎP\{0}) implies a1 … ai-1 ai+1 … anÎP, for some iÎ{1, …, n}; (m, n³2).In this paper several results concerning (n - 1, n)-Fm-prime and (n - 1, n)-weakly prime ideals are proved. We show that in a Noetherian domain a Fm-prime ideal is primary and we show that in some well-known rings (n - 1, n)-Fm-prime ideals and (n - 1, n)-prime ideals coincide.