In this paper, we study some properties of φ-prime submodules andwe give another charactrization for it. For given N and K a moduleM with ⊆ N, prove that is submodule if only N/Kis φ_K-prime submodule. Finally, show any finite sum isφ-prime.

The notion of intuitinitic (T, S)-fuzzy submodule in Module is introduced, and related properties are investigated. Mathematics Subject Classification: 08A72, 13C99

let $r$ be a commutative ring with identity and let $m$ be an $r$-module. a proper submodule $p$ of $m$ is called strongly prime submodule if $(p + rx : m)ysubseteq p$ for $x, yin m$, implies that $xin p$ or $yin p$. in this paper, we study more properties of strongly prime submodules. it is shown that a finitely generated $r$-module $m$ is artinian if and only if $m$ is noetherian and every st...

In this article, we formalize a free Z-module and its property. In particular, we formalize the vector space of rational field corresponding to a free Z-module and prove formally that submodules of a free Z-module are free. Z-module is necassary for lattice problems LLL (Lenstra, Lenstra and Lovász) base reduction algorithm and cryptographic systems with lattice [20]. Some theorems in this arti...

In this paper we characterize the radical of an arbitrary‎ ‎submodule $N$ of a finitely generated free module $F$ over a‎ ‎commutatitve ring $R$ with identity‎. ‎Also we study submodules of‎ ‎$F$ which satisfy the radical formula‎. ‎Finally we derive‎ ‎necessary and sufficient conditions for $R$ to be a‎ ‎Pr$ddot{mbox{u}}$fer domain‎, ‎in terms of the radical of a‎ ‎cyclic submodule in $Rbigopl...

In this paper, we consider the problem of extending existing submodule construction techniques that have been developed for finite state models into more expressive and compact behavioral models that handle data through parameterized interactions, state variables and simple guards. We provide a behavioral model based on extended Input-Output Automata and describe an algorithm that provides the ...

This paper describes the electro-thermal design of a medium voltage modular multilevel converter prototype, from the submodule power semiconductor thermal requirements to the overall system level integration. The high number of semiconductors and capacitors involved are stressed differently depending on the actual operating conditions, as discussed in the paper. The presented design considers a...

Let R be a commutative ring with identity and let M be an infinite unitary R-module. (Unless indicated otherwise, all rings are commutative with identity 1 ̸ = 0 and all modules are unitary.)ThenM is called a Jónsson module provided every proper submodule of M has smaller cardinality than M. Dually, M is said to be homomorphically smaller (HS for short) if |M/N| < |M| for every nonzero submodule...