Normalizing Extensions of Semiprime Rings (original) (raw)

On Centrally Semiprime Rings and Centrally Semiprime

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.

A note on intermediate normalising extensions

Bulletin of the Australian Mathematical Society, 1994

We prove that the following ring-theoretic properties are shared by the two rings involved in a normalising extension R ⊂ S, and that these properties are inherited by any intermediate extension: semilocal, left perfect, semiprimary. This transfer fails for the nilpotency of the Jacobson radical. However, if the normalising set is a basis for the left R-module S, then the nilpotency of the Jacobson radical behaves in the same way as the three properties mentioned above.

On Centrally Prime and Centrally Semiprime Rings

AL-Rafidain Journal of Computer Sciences and Mathematics, 2008

In this paper, centrally prime and centrally semiprime rings are defined and the relations between these two rings and prime (resp. semiprime) rings are studied.Among the results of the paper some conditions are given under which prime (resp. semiprime) rings become centrally prime (resp.centrally semiprime) as in:1-A nonzero prime (resp. semiprime) ring which has no proper zero divisors is centrally prime (resp.centrally semiprime).Also we gave some other conditions which make prime (resp. semiprime) rings and centrally prime (resp.centrally semiprime) rings equivalent, as in :2-A ring which satisfies the-) (BZP for multiplicative systems is prime (resp. semiprime) if and only if it is centrally prime (resp.centrally semiprime).3-A ring with identity in which every nonzero element of its center is a unit is prime (resp. semiprime) if and only if it is centrally prime (resp.centrally semiprime).

S − k−prime and S − k−semiprime ideals of semirings

Discussiones Mathematicae General Algebra and Applications, 2024

Let R be a commutative ring and S a multiplicatively closed subset of R. Hamed and Malek [7] defined an ideal P of R disjoint with S to be an S-prime ideal of R if there exists an s ∈ S such that for all a, b ∈ R if ab ∈ P, then sa ∈ P or sb ∈ P. In this paper, we introduce the notions of S-k-prime and S-k-semiprime ideals of semirings, S-k-m-system, and S-k-p-system. We study some properties and characterizations for S-k-prime and S-k-semiprime ideals of semirings in terms of S-k-m-system and S-kp-system respectively. We also introduce the concepts of S-prime semiring and S-semiprime semiring and study the characterizations for S-k-prime and S-k-semiprime ideals in these two semirings.

Results on multiplicative semiderivations in semiprime rings

International Journal of Algebra

Let R be a semiprime ring. An additive mapping d : R → R is called a semiderivation if there exists a function g : R → R such that (i) d(xy) = d(x)g(y) + xd(y) = d(x)y + g(x)d(y) and (ii) d(g(x)) = g(d(x)) hold for all x, y ∈ R. The aim of this paper is to explore the commutativity of semiprime rings admitting multiplicative semiderivations.

On symmetric biadditive mappings of semiprime rings

Boletim da Sociedade Paranaense de Matemática, 2015

Let R be a ring with centre Z(R). A mapping D(., .) : R× R −→ R issaid to be symmetric if D(x, y) = D(y, x) for all x, y ∈ R. A mapping f : R −→ Rdefined by f(x) = D(x, x) for all x ∈ R, is called trace of D. It is obvious thatin the case D(., .) : R × R −→ R is a symmetric mapping, which is also biadditive(i.e. additive in both arguments), the trace f of D satisfies the relation f(x + y) =f(x) + f(y) + 2D(x, y), for all x, y ∈ R. In this paper we prove that a nonzero left idealL of a 2-torsion free semiprime ring R is central if it satisfies any one of the followingproperties: (i) f(xy) ∓ [x, y] ∈ Z(R), (ii) f(xy) ∓ [y, x] ∈ Z(R), (iii) f(xy) ∓ xy ∈Z(R), (iv) f(xy)∓yx ∈ Z(R), (v) f([x, y])∓[x, y] ∈ Z(R), (vi) f([x, y])∓[y, x] ∈ Z(R),(vii) f([x, y])∓xy ∈ Z(R), (viii) f([x, y])∓yx ∈ Z(R), (ix) f(xy)∓f(x)∓[x, y] ∈ Z(R),(x) f(xy)∓f(y)∓[x, y] ∈ Z(R), (xi) f([x, y])∓f(x)∓[x, y] ∈ Z(R), (xii) f([x, y])∓f(y)∓[x, y] ∈ Z(R), (xiii) f([x, y])∓f(xy)∓[x, y] ∈ Z(R), (xiv) f([x, y])∓f(xy)∓[y, x] ...

LEFT CENTRALIZERS OF SEMIPRIME RINGS

ijpam.eu

Abstract: In this paper we show that a mapping T of a semiprime ring R into itself is a centralizer if and only if it is a centralizing left centralizer. We also prove that if T and S are left centralizers of a semiprime ring R satisfying T(x)x + xS(x) ∈ Z(R) (the center of R) for all x ∈ R, then ...

On Strongly Semi Prime over Noetherian Regular Delta Near Rings and their Extensions(SSPNR-delta-NR)

A Noetherian regular δ-near-ring N with 1 is called an Strongly Semi Prime Noetherian regular δ-near-ring (SSP-NR- δ-NR) if every essential ideal is insulated. As a Semi Prime is equivalent to every essential ideal being faithful then we can see that SSP-NR- δ-NR is Semi Prime. For a Noetherian regular δ-near-ring N with 1 , we show that every proper kernel Functor generates a proper torsion radical if and only if the Noetherian regular δ-near-ring N is a finite sub direct product of strongly prime also called ATF Noetherian regular δ-near-ring N. This is equivalent to every essential right ideal containing a finite set whose right annihilator is zero. We use this characterization to quickly prove a number of properties of Noetherian regular δ-near-ring N of Near Rings say N satisfying this condition and apply the results to the problem : “When is every kernel functor a torsion radical in a Noetherian regular δ-near-ring N”.