In this paper, all rings are associative with identity and all modules are unital left modules unless otherwise specified. Let R be a ring and M a module. N ≤M will mean N is a submodule of M. A submodule E of M is called essential in M (notation E ≤e M) if E∩A = 0 for any nonzero submodule A of M. Dually, a submodule S of M is called small in M (notation S M) if M = S+T for any proper submodul...

In this paper branching rules for the fundamental representations of the symplectic groups in positive characteristic are found. The submodule structure of the restrictions of the fundamental modules for the group Sp2n(K) to the naturally embedded subgroup Sp2n−2(K) is determined. As a corollary, inductive systems of fundamental representations for Sp∞(K) are classified. The submodule structure...

This paper addresses the problem of designing a submodule of a given system ofcommunicating I/O automata. The problem may be formulated mathematically by theequation (C||X)rA under the constraint IX=In, where C represents the specification of theknown part of the system, called the context, A represents the specification of the whole system,X represents the specification of the ...

One of the sciences used in digital security systems is cryptography. Cryptography closely related to integer system, especially prime numbers. Prime numbers themselves have been abstracted a lot. form abstraction ideal. Previous studies proven that an Ideal said be ideal on if and only I constructed by element. Other also shown how develops. them research result Dauns, where developed submodul...

We consider the following problem: For a system consisting of two components, the behavior of one component is known as well as the desired global behavior. What should be the behavior of the second component such that the behavior of the composition of the two conforms to the desired behavior ? This problem has been called "submodule construction" or "equation solving”. Solutions to this probl...

let $r$ be an arbitrary ring and $t$ be a submodule of an $r$-module $m$. a submodule $n$ of $m$ is called $t$-small in $m$ provided for each submodule $x$ of $m$, $tsubseteq x+n$ implies that $tsubseteq x$. we study this mentioned notion which is a generalization of the small submodules and we obtain some related results.

Page Introduction 269 Notation and terminology 271 I. Basic facts 271 1. Review of IM 271 2. Review of AM 273 3. Solvable groups, aG, finite Priifer rank, FP 2 , and HNN-extensions . 276 4. Matrix computation of AM in the finite rank case . . . . 277 5. Finiteness conditions: F P m and finite presentation . . . . 278 6. The properties m-tameness and w-domestication 280 II. Proof that Hm{G, Y\ R...