This paper presents the concept of the notions of bipolar fuzzy commutative ideals of a BCK-algebra. We also study various properties regarding this concept. We prove various characterisations for bipolar fuzzy commutative ideals. Moreover, a relationship between bipolar fuzzy commutative ideals and bipolar fuzzy positive implicative ideals is presented. In addition, we present the notation of bipolar fuzzy characteristic commutative ideal. Then a relationship between a bipolar fuzzy characteristic commutative ideal and its level-cut commutative ideals is provided.

In this paper, we introduce the notion of a pythagorean fuzzy small submodule. We prove various characterisations for pythagorean fuzzy small submodules. We provide a relation between a pythagorean fuzzy small submodule and a basic small submodule. In addition, some important properties regarding pythagorean fuzzy small submodules are investigated.

In this paper, the notation of interior KU-algebra and some related definitions are introduced. Various characterisations for the interior KU-algebra have been proved. The concept of interior positive implicative KU-algebra and prove some results regarding this concept are also presented. In addition, the interior KU-ideals and several properties of these ideals are presented.

This paper investigates cyclic codes over $F_q^n\times P^m\times Q^k$, where F_q is a finite field of odd characteristic, $P=F_q\left[v\right]/⟨v^2-{\tau }_{2}v⟩$, and $Q=F_q[u,v]/⟨u^3-{\tau }_1^{2}u,v^2-{\tau }_{2}v⟩$, with ${\tau }_1,{\tau }_2\in F_q^{*}$. Using complete systems of orthogonal idempotents, we decompose (P) and (Q) into two and six constituent codes over F_q, respectively, and define matrix-based Gray maps. The induced Gray map from $F_q^n\times P^m\times Q^k$ to $F_q^{n+2m+6k}$ is shown to be F_q-linear and Lee–Hamming distance preserving. We prove that a mixed-alphabet code is cyclic if and only if its deinterleaved Gray image is a generalized quasi-cyclic code with block-length profile (n, m, m, k, k, k, k, k, k). A polynomial-module characterization, constituent generator polynomials, dimension bounds, shift-generated spanning sets, and structural results for the Euclidean dual and hull are also established. Furthermore, the generator structure and cardinality of cyclic codes over P, Q, and the mixed alphabet are determined through their field constituents. As an application, scalar-orthogonal Gray matrices and dual-containing constituent cyclic codes are combined with the Calderbank–Shor–Steane construction. An explicit construction over F₅ yields a dual-containing generalized quasi-cyclic code with parameters [90, 75, 3]₅, and a quantum error-correcting code with parameters [[90, 60, 3]]₅.