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In algebraic theory, there are groups, rings, modules over a ring, and algebra over rings. As a result, people try to develop theories that apply to groups to apply to other concepts. For example, Goursat's theorem in groups related to the direct product of the two groups, then developed in the ring, the module over a ring, the algebra over a ring related to the direct product of the two rings, two modules over a ring, and two algebras over a ring, respectively. Can we extend Goursat's theorem in algebras over a ring R (R-algebras) with respect to the direct product of n R-algebras? So, this paper will extend Goursat's theorem to direct products of n R-algebras, exploring related properties and providing formal proof. The main result of this study demonstrates that every subalgebra in a direct product of n R-algebras can be uniquely determined by n − 1 R-algebra epimorphisms of an algebra to a factor algebra. [ABSTRACT FROM AUTHOR]

In [1], Faco and Marques gave the conditions that even perfect numbers belonging to k-generalized Fibonacci numbers. In this paper, we present alternative proof for being even perfect numbers in Tribonacci sequence. Our methods depends on p-adic order of a Tribonacci number. Moreover, we present the perfect numbers with distance 1 to Tribonacci numbers. [ABSTRACT FROM AUTHOR]

This paper presents hourglass matrix, a dense square matrix which is obtained from Quadrant Interlocking Factorization (QIF) of nonsingular matrix. We discuss the condition to generate its nonzero entries and formulate its total number of zero and nonzero entries. Furthermore, the determinant of even and odd order of the matrix as well as the eigenvalue of the odd order of the matrix are examined. In all, we give some deductions to differentiate between hourglass matrix and classical Z-matrix. [ABSTRACT FROM AUTHOR]

We show some results about the existence of regular factors in star-free graphs with large connectivity. [ABSTRACT FROM AUTHOR]

This paper aims to propose a new design that is called an array cyclic design. In this developing this design, a combination between a strictly simple cyclic triple system (SSCTS (v, ƛ)) and near-two-factor will be constructed. In doing so, we need to introduce new factors namely a cycle factor and a wheel factor. Then we present an analysis for case v = 8. [ABSTRACT FROM AUTHOR]