On Graphs Associated With Modules Over Commutative Rings

Abstract

Abstract. Let M be an R-module, where R is a commutative ring with identity 1 and let G(V, E) be a graph. In this paper, we study the graphs associated with modules over commutative rings. We associate three simple graphs ann f (Γ(M R )), anns(Γ(M R )) and annt(Γ(M R )) to M called full annihilating, semi-annihilating and star-annihilating graph. When M is finite over R, we investigate metric dimensions in ann f (Γ(M R )), anns(Γ(M R )) and annt(Γ(M R )). We show that M over R is finite if and only if the metric dimension of the graph ann f (Γ(M R )) is finite. We further show that the graphs ann f (Γ(M R )), anns(Γ(M R )) and annt(Γ(M R )) are empty if and only if M is a prime-multiplicationlike R-module. We investigate the case when M is a free R-module, where R is an integral domain and show that the graphs ann f (Γ(M R )), anns(Γ(M R )) and annt(Γ(M R )) are empty if and only if M ∼ = R. Finally, we characterize all the non-simple weakly virtually divisible modules M for which Ann(M ) is a prime ideal and Soc(M ) = 0.

On the metric dimension of strongly annihilating-ideal graphs of commutative rings

“…Recently, there was much work done in computing the metric dimension of graphs associated with algebraic structures. Calculating the metric dimension for the commuting graph of a dihedral group was done in [1], for the zero-divisor graphs of commutative rings in [9,10,12], for the compressed zero-divisor graphs of commutative rings in [13], for total graphs of finite commutative rings in [6], for some graphs of modules in [11] and for annihilator graphs of commutative rings in [15]. Motivated by these papers, we study the metric dimension of another graph associated with a commutative ring.…”

Computing metric dimension of compressed zero divisor graphs associated to rings

“…Anderson and Livingston [3] studied zero divisor graph of non-zero zero divisors of a commutative ring R. For a commutative ring R with 1 = 0, let Z * (R) = Z(R) \ {0} be the set of non-zero zero divisors of R. A zero divisor graph Γ (R) is the undirected graph with vertex set Z * (R) and the two vertices x and y are adjacent if and only if xy = 0. This zero divisor graph has been studied extensively and even more the idea has been extended to the ideal based zero divisor graphs in [15,23] and modules in [20]. Inspired by ideas from Mulay [16], we study the zero divisor graph of equivalence classes of zero divisors of a ring R. Anderson and LaGrange [4] ).…”

On combinatorial aspects of modules over commutative rings

“…The combinatorial properties of zero-divisors discovered in [7] has also been studied in module theory. Recently in [19], the elements of a module M has been classified into full-annihilators, semi annihilators and star-annihilators, see Definition 2.1 in section 2. For M = R, these elements are the zero-divisors of a ring R, so the three simple graphs ann f (Γ(M )), ann s (Γ(M )) and ann t (Γ(M )) corresponding to full-annihilators, semi annihilators and star-annihilators in M are natural generalizations of a zero-divisor graph introduced in [1].…”

“…In [19] authors introduced annihilating graphs arising from modules over commutative rings called as full-annihilating, semi-annihilating and star-annihilating graphs denoted by ann f (Γ(M )), ann s (Γ(M )) and ann t (Γ(M )) respectively. The vertices of annihilating graphs are elements of objects A f (M ), A s (M ) and A t (M ), and two vertices x and y are adjacent if and only if [x : M ][y : M ]M = 0.…”

“…But an infinite intersection of essential ideals need not to be an essential ideal, even a countable intersection of essential ideals in general is not an essential ideal as can be seen in [6]. If ann f (Γ(M )) is a finite graph, then by [Theorem 3.3, [19]] M is finite over R, so the submodules determined by the vertices of graph are finite and therefore the ideals corresponding to submodules are finite in number and we conclude the intersection of essential ideals [x : M ], x ∈ A f (M ) in R is an essential ideal. Motivated by [6], we have have the following question regarding essential ideals corresponding to submodules M determined by vertices of the graph ann f (Γ(M )).…”