# Domination number in the annihilating-ideal graphs of commutative rings

## Summary (1 min read)

### 1. Introduction

- The study of algebraic structures, using the properties of graphs, have become an exciting research topic in the past twenty years, leading to many interesting results and questions.
- Throughout this paper, all rings are assumed to be commutative rings with identity.
- A subset 2010 Mathematics Subject Classification: Primary 13A15; Secondary 05C75.
- Also, some relations between the domination numbers of annihilating-ideal graphs and zero-divisor graphs are studied.

### 2. Main results

- The authors start with the following remark which completely characterizes all rings for which either γ(AG(R)) = 1 or γ(Γ(R)) = 1. Remark 2.1.
- Now, the authors can state the following proposition.
- The following theorem provides an upper bound for the domination number of the annihilating-ideal graph of a Noetherian ring.

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### Cites background from "Domination number in the annihilati..."

...Also, we note a mistake of [28] and we characterize the domination of a ring in which the zero ideal is a fixed-place ideal and domination of AG(X) in which X is almost discrete and finally we prove that dt(AG(R)) is finite, if and only if dtt(AG(R)) is finite; if and only if Min(R) is finite....

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...Also, First the annihilatingideal graph has been introduced and studied in [19] and then it has been studied in several articles; see [20, 9, 2, 1, 27, 22, 28]....

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##### References

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### "Domination number in the annihilati..." refers background in this paper

...There are many papers on assigning a graph to a ring, for instance see [1, 3, 9, 11]....

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956 citations

### "Domination number in the annihilati..." refers background in this paper

...The concept of the zero-divisor graph was first introduced by Beck (see [5])....

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302 citations

##### Related Papers (5)

##### Frequently Asked Questions (13)

###### Q2. What is the condition of R to be an Artinian ring?

Since R ≇ F1 × F2, where F1 and F2 are two fields, the authors find that Jins = Jint = 0, for some i, t, s, where 1 6 i 6 k − 1 and 1 6 t, s 6 k.

###### Q3. What is the annihilating-ideal graph of R?

The zero-divisor graph of R, Γ(R), is a graph with the vertex set Z(R)r {0} and two distinct vertices x and y are adjacent if and only if xy = 0.

###### Q4. What is the prime ideal of a commutative ring?

A prime ideal p is said to be an associated prime ideal of a commutative Noetherian ring R, if there exists a nonzero element x in R such that p = ann(x).

###### Q5. What is the remark that describes the relation between t(AG(R)?

By [6, Theorem 2.2], there is a vertex of AG(R) which is adjacent to every other vertex if and only if either R ∼= F × D, where F is a field and D is an integral domain, or Z(R) is an annihilator ideal.

###### Q6. what is the remark that describes the relation between (R) and ?

by [3, Theorem 2.5], there is a vertex of Γ(R) which is adjacent to every other vertex if and only if either R ∼= Z2 × D, where D is an integral domain, or Z(R) is an annihilator ideal.

###### Q7. What is the proof of the annihilating-ideal graph of a fi?

The authors may assume that γ(AG(R1)) = m and γ(AG(R2)) = n, for some positive integers m and n. Let {I1, . . . , Im} and {J1, . . . , Jn} be two minimal dominating sets in AG(R1) and AG(R1), respectively.

###### Q8. what is the inverse of the ring?

For a ring R, which is a product of two (nonzero) rings, one of the following holds:(i) If R ∼= F × D, where F is a field and D is an integral domain, then γ(AG(R)) = 1.(ii) If R ∼= D1 × D2, where D1 and D2 are integral domains which are not fields, then γ(AG(R)) = 2.(iii)

###### Q9. what is the inverse of the ag?

Since n > 3, the ideals pi and pj , for i 6= j are not adjacent (from pipj = 0 ⊆ pk it would follow that pi ⊆ pk, or pj ⊆ pk which is not true).

###### Q10. What is the domination number of a annihilating-ideal graph?

For reduced rings with finitely many minimal primes and Artinian rings, the domination number of the annihilating-ideal graphs is given.

###### Q11. What is the inverse of the inverse of the inverse?

With no loss of generality, one can assume that γ(AG(R1)) < ∞. Suppose that γ(AG(R1)) = n and {I1, . . . , In} is a minimal dominating set of AG(R1).

###### Q12. What is the meaning of a graph?

The study of algebraic structures, using the properties of graphs, have become an exciting research topic in the past twenty years, leading to many interesting results and questions.

###### Q13. What is the sum of all prime ideals of R?

By Ass(R) the authors denote the set of all associated prime ideals of R. A ring R is said to be reduced, if it has no nonzero nilpotent element or equivalently ⋂