The generalized total graph of modules respect to proper submodules over commutative rings.

Document Type : Research Paper

Authors

¹ Islamic Azad University

² University of Tehran

Abstract

Let

M

be a module over a commutative ring

R

and let

N

be a proper submodule of

M

. The total graph of

M

over

R

with respect to

N

, denoted by

T (Γ_{N} (M))

, have been introduced and studied in [2]. In this paper, A generalization of the total graph

T (Γ_{N} (M))

, denoted by

T (Γ_{N, I} (M))

is presented, where

I

is an ideal of

R

. It is the graph with all elements of

M

as vertices, and for distinct

m, n \in M

, the vertices

m

and

n

are adjacent if and only if

m + n \in M (N, I)

, where

M (N, I) = {m \in M : r m \in N + I M f o r s o m e r \in R - I}

. The main purpose of this paper is to extend the definitions and properties given in [2] and [12] to a more general case.

Keywords