in this article, we give several generalizations of the concept of annihilating ideal graph over a commutative ring with identity to modules. weobserve that over a commutative ring $r$, $bbb{ag}_*(_rm)$ isconnected and diam$bbb{ag}_*(_rm)leq 3$. moreover, if $bbb{ag}_*(_rm)$ contains a cycle, then $mbox{gr}bbb{ag}_*(_rm)leq 4$. also for an $r$-module $m$ with$bbb{a}_*(m)neq s(m)setminus {0}$, $...

An elementary proof is given for Hutchinson’s duality theorem, which states that if a lattice identity λ holds in all submodule lattices of modules over a ring R with unit element then so does the dual of λ.

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 introduce a new class of modules which is closely related to the class of Noetherian modules. Let $R$ be a commutative ring with identity and let $M$ be an $R$-module such that $Nil(M)$ is a divided prime submodule of $M$. $M$ is called a Nonnil-Noetherian $R$-module if every nonnil submodule of $M$ is finitely generated. We prove that many of the properties of Noetherian modul...

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...

This paper extends the theory of the Gröbner fan and Gröbner walk for ideals in polynomial rings to the case of submodules of free modules over a polynomial ring. The Gröbner fan for a submodule creates a correspondence between a pair consisting of a cone in the fan and a point in the support of the cone and a pair consisting of a leading monomial submodule (or equivalently, a reduced marked Gr...

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