Abstract. In a multiplication module, prime submodules have the property, if a prime submodule contains a finite intersection of submodules then one of the submodules is contained in the prime submodule. In this paper, we generalize this property to infinite intersection of submodules and call such prime submodules strongly prime submodule. A multiplication module in which every prime submodule is strongly prime will be called strongly 0-dimensional module. It is also an extension of strongly 0-dimensional rings. After this we investigate properties of strongly 0-dimensional modules and give relations of von Neumann regular modules, Qmodules and strongly 0-dimensional modules.

In this work, we add an additional condition to strong pseudo prime test to base 2. Then, we provide theoretical and heuristics evidences showing that the resulting algorithm catches all composite numbers. Our method is based on the structure of singular cubics' Jacobian groups on which we also define an effective addition algorithm.

In this work, we introduce the notion of S-1-absorbing primary submodule as an extension of 1-absorbing primary submodule. Let S be a multiplicatively closed subset of a ring R and M be an R-module. A submodule N of M with (N : R M )∩S = ∅ is said to be S-1-absorbing primary if whenever abm ∈ N for some non-unit a, b ∈ R and m ∈ M , then either sab ∈ (N : R M ) or sm ∈ M -rad(N ). We examine several properties of this concept and provide some characterizations. In addition, S-1-absorbing primary avoidance theorem and S-1-absorbing primary property for idealization and amalgamation are presented.

In this paper, we define coprimely structured rings, which are the generalization of strongly 0-dimensional rings. Furthermore, we investigate coprimely structured rings and give some relations between other rings such as Artinian rings, strongly 0-dimensional rings, and h-local domains.

Sharing a secret key between two physically separated nodes, Alice and Bob, is possible through the use of quantum key distribution (QKD) techniques. In the presence of an eavesdropper, Alice's key may not be identical with Bob's key, due to the characteristics of a quantum channel. To obtain identical keys at Alice and Bob, we propose a block-based key verification protocol that relies on Newton's polynomial interpolation. As the nodes solely share random numbers and indices of the removed blocks, no information is revealed about the secret message, at a cost of higher computational complexity. The error propagation through the key verification process is prevented by the characteristics of the proposed approach.