On strongly prime rings and ideals

Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 2016

Let R be a commutative ring with identity 1 = 0 and let I be a proper ideal of R. D. D. Anderson and E. Smith called I weakly prime if a, b ∈ R and 0 = ab ∈ I implies a ∈ I or b ∈ I. In this paper, we define I to be weakly semiprime if a ∈ R and 0 = a 2 ∈ I implies a ∈ I. For example, every proper ideal of a quasilocal ring (R, M) with M 2 = 0 is weakly semiprime. We give examples of weakly semiprime ideals that are neither semiprime nor weakly prime. We show that a weakly semiprime ideal of R that is not semiprime is a nil ideal of R. We show that if I is a weakly semiprime ideal of R that is not semiprime and 2 is not a zero-divisor of of R, then I 2 = {0} (and hence i 2 = 0 for every i ∈ I). We give an example of a ring R that admits a weakly semiprime ideal I that is not semiprime where i 2 = 0 for some i ∈ I. If R = R 1 × R 2 for some rings R 1 , R 2 , then we characterize all weakly semiprime ideals of R that are not semiprime. We characterize all weakly semiprime ideals of of Z m that are not semiprime. We show that every proper ideal of R is weakly semiprime if and only if either R is von Neumann regular or R is quasilocal with maximal ideal Nil(R) such that w 2 = 0 for every w ∈ Nil(R). Keywords Primary ideal • Prime ideal • Weakly prime ideal • 2-absorbing ideal • n-absorbing ideal • Semiprime • Weakly semiprime ideal

Mediterranean Journal of Mathematics, 2012

In this paper, we introduce a strong property (A) as follows: A ring R is called satisfying strong property (A) if every finitely generated ideal of R which is generated by a finite number of zero-divisors elements of R, has a non zero annihilator. We study the transfer of property (A) and strong property (A) in trivial ring extensions and amalgamated duplication of a ring along an ideal. We also exhibit a class of rings which satisfy property (A) and do not satisfy strong property (A).

São Paulo Journal of Mathematical Sciences, 2020

In this paper, we study WP-ring, that is a ring in which every nonzero weakly prime ideal is prime. Some results including the characterizations and the transfer of WPring property to homomorphic image and localization are given. Also, we study the possible transfer of the WP-ring property between A and A ∝ E and between A and A ⋈ f J. Our results provide new classes of commutative rings satisfying the above property.

2016

Some special classes μ whose upper radical U(μ) determined by μ coincides with the upper radical U(* μ) k determined by (* μ) k

Journal of the Australian Mathematical Society, 1983

It is well-known that in any near-ring, any intersection of prime ideals is a semi-prime ideal. The aim of this note is to prove that any ideal is a prime ideal if and only if it is equal to its prime radical. As a consequence of this we have any semi-prime ideal I in a near-ring N is the intersection of minimal prime ideals of I in N and that I is the intersection of all prime ideals containing I.

Maǧallaẗ ǧāmiʻaẗ kirkūk, 2008

In this paper, two new algebraic structures are introduced which we call a centrally semiprime ring and a centrally semiprime right near-ring, and we look for those conditions which make centrally semiprime rings as commutative rings, so that several results are proved, also we extend some properties of semiprime rings and semiprime right near-rings to centrally semiprime rings and centrally semiprime right near-rings.

Communications in Algebra, 2003

It is well known that a domain without proper strongly divisorial ideals is completely integrally closed. In this paper we show that a domain without prime strongly divisorial ideals is not necessarily completely integrally closed, although this property holds under some additional assumptions.

Bulletin of The Australian Mathematical Society, 1993

Quaestiones Mathematicae, 2020

Let A be an integral domain with quotient field K. A. Badawi and E. Houston called a strongly primary ideal I of A if whenever x, y ∈ K and xy ∈ I, we have x ∈ I or y n ∈ I for some n ≥ 1. In this note, we study the generalization of strongly primary ideal to the context of arbitrary commutative rings. We define a primary ideal P of A to be strongly primary if for each a, b ∈ A, we have aP ⊆ bA or b n A ⊆ a n P for some n ≥ 1.

Journal of Pure and Applied Algebra, 2003

A unitary strongly prime ring is defined as a prime ring whose central closure is simple with identity element. The class of unitary strongly prime rings is a special class of rings and the corresponding radical is called the unitary strongly prime radical. In this paper we prove some results on unitary strongly prime rings. The results are applied to study the unitary strongly prime radical of a polynomial ring and also R-disjoint maximal ideals of polynomial rings over R in a finite number of indeterminates. From this we get relations between the Brown-McCoy radical and the unitary strongly prime radical of polynomial rings. In particular, the Brown-McCoy radical of R[X] is equal to the unitary strongly prime radical of R[X] and also equal to S(R)[X], where S(R) denotes the unitary strongly prime radical of R, when X is an infinite set of either commuting or non-commuting indeterminates. For a PI ring R this holds for any set X.

International Journal of Mathematics and Mathematical Sciences, 1994

LetRbe a ring, and letCdenote the center ofR.Ris said to have a prime center if wheneverabbelongs toCthenabelongs toCorbbelongs toC. The structure of certain classes of these rings is studied, along with the relation of the notion of prime centers to commutativity. An example of a non-commutative ring with a prime center is given.

2021

Let R be a prime ring with the extended centroid C and the Matrindale quotient ring Q. An additive mapping F : R → R is called a semiderivation associated with a mapping G : R → R, whenever F (xy) = F (x)G (y) + xF (y) = F (x)y + G (x)F (y) and F (G (x)) = G (F (x)) holds for all x, y ∈ R. In this manuscript, we investigate and describe the structure of a prime ring R which satisfies F (x ◦ y) ∈ Z (R) for all x, y ∈ R, where m,n ∈ Z and F : R → R is a semiderivation with an automorphism ξ of R. Further, as an application of our ring theoretic results, we discussed the nature of C ∗-algebras. To be more specific, we obtain for any primitive C ∗-algebra A . If an anti-automorphism ζ : A → A satisfies the relation (x) +x ∈ Z (A ) for every x, y ∈ A , then A is C ∗ −W4-algebra, i. e., A satisfies the standard identity W4(a1, a2, a3, a4) = 0 for all a1, a2, a3, a4 ∈ A .

Journal of Mathematical and Fundamental Sciences

In this paper, assuming that is a near-ring and is an ideal of , the-center of , the-center of an element in , the-identities of are defined. Their properties and relations are investigated. It is shown that the set of allidentities in is a multiplicative subsemigroup of. Also,-right and-left permutable and-medial near-rings are defined and some properties and connections are given.-regular and-strongly regular near-rings are studied.-completely prime ideals are introduced and some characterizations ofcompletely prime near-rings are provided. Also, some properties ofidempotents,-centers,-identities incompletely prime near-rings are investigated. The results that were obtained in this study are illustrated with many examples.

Journal of Algebra, 2017

The purpose of the paper is to study prime rings R such that the central closure RC is a simple ring with 1 and it is finitely generated over R by elements of the extended centroid C, that is, RC = R[c 1 ,. .. , c n ], for some c 1 ,. .. , c n ∈ C. In particular, we will show that if there exists a prime ring with zero center whose central closure is simple with 1 and generated by finitely many central elements, then there exists such a ring whose central closure is generated by two central elements.

Journal of the Australian Mathematical Society, 2008

A proper ideal I of a ring R is said to be strongly irreducible if for each pair of ideals A and B of R, A ∩ B ⊆ I implies that either A ⊆ I or B ⊆ I . In this paper we study strongly irreducible ideals in different rings. The relations between strongly irreducible ideals of a ring and strongly irreducible ideals of localizations of the ring are also studied. Furthermore, a topology similar to the Zariski topology related to strongly irreducible ideals is introduced. This topology has the Zariski topology defined by prime ideals as one of its subspace topologies.