Ayman Badawi

On 2-absorbing ideals of commutative rings

Suppose that R is a commutative ring with 1 ≠ 0. In this paper, we introduce the concept of 2-absorbing ideal which is a generalisation of prime ideal. A nonzero proper ideal I of R is called a 2-absorbing ideal of R if whenever a, b, c ∈ R and abc ∈ I , then ab ∈ I or ac ∈ I or bc ∈ I . It is shown that a nonzero proper ideal I of R is a 2-absorbing ideal if and only if whenever I 1 I 2 I 3 ⊆ I for some ideals I 1 , I 2 , I 3 of R , then I 1 I 2 ⊆ I or I 2 I 3 ⊆ I or I 1 I 3 ⊆ I . It is shown that if I is a 2-absorbing ideal of R , then either Rad( I ) is a prime ideal of R or Rad( I ) = P 1 ⋂ P 2 where P 1 , P 2 are the only distinct prime ideals of R that are minimal over I . Rings with the property that every nonzero proper ideal is a 2-absorbing ideal are characterised. All 2-absorbing ideals of valuation domains and Prufer domains are completely described. It is shown that a Noetherian domain R is a Dedekind domain if and only if a 2-absorbing ideal of R is either a maximal ideal of R or M 2 for some maximal ideal M of R or M 1 M 2 where M 1 , M 2 are some maximal ideals of R . If R M is Noetherian for each maximal ideal M of R , then it is shown that an integral domain R is an almost Dedekind domain if and only if a 2-absorbing ideal of R is either a maximal ideal of R or M 2 for some maximal ideal M of R or M 1 M 2 where M 1 , M 2 are some maximal ideals of R .

On the Zero-Divisor Graph of a Ring

Let R be a commutative ring with identity, Z(R) its set of zero-divisors, and Nil(R) its ideal of nilpotent elements. The zero-divisor graph of R is Γ(R) = Z(R)\{0}, with distinct vertices x and y adjacent if and only if xy = 0. In this article, we study Γ(R) for rings R with nonzero zero-divisors which satisfy certain divisibility conditions between elements of R or comparability conditions between ideals or prime ideals of R. These rings include chained rings, rings R whose prime ideals contained in Z(R) are linearly ordered, and rings R such that {0} ≠ Nil(R) ⊆ zR for all z ∈ Z(R)\Nil(R).

On n-Absorbing Ideals of Commutative Rings

Let R be a commutative ring with 1 ≠ 0 and n a positive integer. In this article, we study two generalizations of a prime ideal. A proper ideal I of R is called an n-absorbing (resp., strongly n-absorbing) ideal if whenever x 1…x n+1 ∈ I for x 1,…, x n+1 ∈ R (resp., I 1…I n+1 ⊆ I for ideals I 1,…, I n+1 of R), then there are n of the x i s (resp., n of the I i s) whose product is in I. We investigate n-absorbing and strongly n-absorbing ideals, and we conjecture that these two concepts are equivalent. In particular, we study the stability of n-absorbing ideals with respect to various ring-theoretic constructions and study n-absorbing ideals in several classes of commutative rings. For example, in a Noetherian ring every proper ideal is an n-absorbing ideal for some positive integer n, and in a Prüfer domain, an ideal is an n-absorbing ideal for some positive integer n if and only if it is a product of prime ideals.

ON 2-ABSORBING PRIMARY IDEALS IN COMMUTATIVE RINGS

Let R be a commutative ring with 1 � . In this paper, we introduce the concept of 2-absorbing primary ideal which is a general- ization of primary ideal. A proper ideal I of R is called a 2-absorbing primary ideal of R if whenever a, b, c ∈ R and abc ∈ I ,t henab ∈ I or ac ∈ √ I or bc ∈ √ I. A number of results concerning 2-absorbing primary ideals and examples of 2-absorbing primary ideals are given.

On divided commutative rings

Let R be a commutative ring with identity having total quotient ring T. A prime ideal P of R is called divided if P is comparable to every principal ideal of R. If every prime ideal of R is divided, then R is called a divided ring. If P is a nonprincipal divided prime, then P-1 = { x ∊ T : xP ⊃ P} is a ring. We show that if R is an atomic domain and divided, then the Krull dimension of R ≤ 1. Also, we show that if a finitely generated prime ideal containing a nonzerodivisor of a ring R is divided, then it is maximal and R is quasilocal.

On the Annihilator Graph of a Commutative Ring

Let R be a commutative ring with nonzero identity, Z(R) be its set of zero-divisors, and if a ∈ Z(R), then let ann R (a) = {d ∈ R | da = 0}. The annihilator graph of R is the (undirected) graph AG(R) with vertices Z(R)* = Z(R)∖{0}, and two distinct vertices x and y are adjacent if and only if ann R (xy) ≠ ann R (x) ∪ ann R (y). It follows that each edge (path) of the zero-divisor graph Γ(R) is an edge (path) of AG(R). In this article, we study the graph AG(R). For a commutative ring R, we show that AG(R) is connected with diameter at most two and with girth at most four provided that AG(R) has a cycle. Among other things, for a reduced commutative ring R, we show that the annihilator graph AG(R) is identical to the zero-divisor graph Γ(R) if and only if R has exactly two minimal prime ideals.

THE GENERALIZED TOTAL GRAPH OF A COMMUTATIVE RING

Let R be a commutative ring with nonzero identity and H be a nonempty proper subset of R such that R\H is a saturated multiplicatively closed subset of R. The generalized total graph of R is the (simple) graph GTH(R) with all elements of R as the vertices, and two distinct vertices x and y are adjacent if and only if x + y ∈ H. In this paper, we investigate the structure of GTH(R).

ON THE TOTAL GRAPH OF A COMMUTATIVE RING WITHOUT THE ZERO ELEMENT

Let R be a commutative ring with nonzero identity, and let Z(R) be its set of zero-divisors. The total graph of R is the (undirected) graph T(Γ(R)) with vertices all elements of R, and two distinct vertices x and y are adjacent if and only if x + y ∈ Z(R). In this paper, we study the two (induced) subgraphs Z0(Γ(R)) and T0(Γ(R)) of T(Γ(R)), with vertices Z(R)\{0} and R\{0}, respectively. We determine when Z0(Γ(R)) and T0(Γ(R)) are connected and compute their diameter and girth. We also investigate zero-divisor paths and regular paths in T0(Γ(R)).

POWERFUL IDEALS, STRONGLY PRIMARY IDEALS, ALMOST PSEUDO-VALUATION DOMAINS, AND CONDUCIVE DOMAINS

ABSTRACT Let R be a domain with quotient field K, and let I be an ideal of R. We say that I is powerful (strongly primary) if whenever and , we have or ( or for some ). We show that an ideal with either of these properties is comparable to every prime ideal of R, that an ideal is strongly primary it is a primary ideal in some valuation overring of R, and that R admits a powerful ideal R admits a strongly primary ideal R is conducive in the sense of Dobbs-Fedder. Finally, we study domains each of whose prime ideals is strongly primary.

On the Dot Product Graph of a Commutative Ring

Let A be a commutative ring with nonzero identity, 1 ≤ n < ∞ be an integer, and R = A × A × … ×A (n times). The total dot product graph of R is the (undirected) graph TD(R) with vertices R* = R∖{(0, 0,…, 0)}, and two distinct vertices x and y are adjacent if and only if x·y = 0 ∈ A (where x·y denote the normal dot product of x and y). Let Z(R) denote the set of all zero-divisors of R. Then the zero-divisor dot product graph of R is the induced subgraph ZD(R) of TD(R) with vertices Z(R)* = Z(R)∖{(0, 0,…, 0)}. It follows that each edge (path) of the classical zero-divisor graph Γ(R) is an edge (path) of ZD(R). We observe that if n = 1, then TD(R) is a disconnected graph and ZD(R) is identical to the well-known zero-divisor graph of R in the sense of Beck–Anderson–Livingston, and hence it is connected. In this paper, we study both graphs TD(R) and ZD(R). For a commutative ring A and n ≥ 3, we show that TD(R) (ZD(R)) is connected with diameter two (at most three) and with girth three. Among other things, for n ≥ 2, we show that ZD(R) is identical to the zero-divisor graph of R if and only if either n = 2 and A is an integral domain or R is ring-isomorphic to ℤ2 × ℤ2 × ℤ2.