Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 18 (2017), No. 1, pp. 17–29 DOI: 10.18514/MMN.2017.1569 AN EXTENSION OF TOTAL GRAPH OVER A MODULE A. ABBASI AND A. RAMIN Received 01 March, 2015 Abstract. Let R be a commutative ring with nonzero identity and U.R/ its multiplicative group of units. Let M be an R-module where the collection of prime submodules is non-empty and let N  be an arbitrary union of prime submodules. Also, suppose that c 2 U.R/ such that c 1 D c . We define the extended total graph of M as a simple graph T c .M;N  / with vertex set M , and two distinct elements x;y 2 M are adjacent if and only if x C cy 2 N  . In this paper, we will study some graph theoretic results of T c .M;N  /. 2010 Mathematics Subject Classification: 05C25; 13C99 Keywords: total graph, prime submodule 1. I NTRODUCTION Let R be commutative ring with 1 ¤ 0, U.R/ its multiplicative group of units and Z.R/ its set of zero-divisors. A proper submodule N of M is said to be a prime submodule if whenever rm 2 N for some r 2 R and m 2 M , then either m 2 N or r 2 .N W R M/. Clearly, if N is a prime submodule of M , then P D .N W R M/ is a prime ideal of R. Let M be an R-module, T.M/ its set of torsion elements and fN  g 2˝ its set of all prime submodules. The R-module M is said to be primeless if ˝ D ¿. For a submodule L of an R-module M , the ideal fr 2 RjrM  Lg and submodule fm 2 M jrm  Lg will be denoted by .L W R M/ and .L W M r/, respectively. Let N  D S 2 N  be a proper subset of M , and let H  D .N  W R M/ for ¿ ¤   ˝. It can be shown that H  D S 2 P  . The total graph of R was introduced by Anderson and Badawi in [4], as the graph with all elements of R as vertices, and two distinct vertices x;y 2 R are adjacent if and only if x C y 2 Z.R/. Also they introduced in [5] the generalized total graph of R in which Z.R/ is extended to H ,a multiplicative  prime subset of R, in such away that ab 2 H for every a 2 H and b 2 R, and whenever ab 2 H for all a;b 2 R, then either a 2 H or b 2 H . In fact, it is easily seen that H is a multiplicative-prime subset of R if and only if R n H is a saturated multiplicatively closed subset of R. Thus H is a multiplicative-prime subset of R if and only if H is a union of prime ideals of R. c  2017 Miskolc University Press