Semi-Prime Modules

Properties and characterizations for prime and semiprime rings have been provided by A. W. Goldie (2, 3). In a previous paper (1), the authors used the results of (2) to characterize prime and uniform prime modules. It is the aim of the present paper to generalize Goldie's work on semi-prime rings (3) to modules. In this setting certain new properties will appear.Notationally, in the work to follow, the symbol R always denotes a ring and all R-modules will be right R-modules.In the theory of rings an ideal C is said to be prime if and only if whenever AB ⊆ C for ideals A and B, then either A ⊆ C or B ⊆ C. A ring is prime if the zero ideal is prime.

Symmetric generalized biderivations on prime rings

Boletim da Sociedade Paranaense de Matemática ◽

10.5269/bspm.40481 ◽

2021 ◽

Vol 39 (4) ◽

pp. 65-72

Author(s):

Faiza Shujat

Keyword(s):

Prime Ring ◽

Prime Rings ◽

The purpose of the present paper is to prove some results concerning symmetric generalized biderivations on prime and semiprime rings which partially extend some results of Vukman \cite {V}. Infact we prove that: let $R$ be a prime ring of characteristic not two and $I$ be a nonzro ideal of $R$. If $\Delta$ is a symmetric generalized biderivation on $R$ with associated biderivation $D$ such that $[\Delta(x,x), \Delta(y,y)]=0$ for all $x,y \in I$, then one of the following conditions hold\\ \begin{enumerate} \item $R$ is commutative. \item $\Delta$ acts as a left bimultiplier on $R$. \end{enumerate}

On Prime and Semiprime Rings with Derivations

Algebra Colloquium ◽

10.1142/s1005386706000320 ◽

2006 ◽

Vol 13 (03) ◽

pp. 371-380 ◽

Cited By ~ 14

Author(s):

Nurcan Argaç

Keyword(s):

Prime Ring ◽

Nonempty Subset ◽

Semiprime Ring ◽

Nonzero Ideal ◽

Prime Rings ◽

Semiprime Rings ◽

Torsion Free

Let R be a ring and S a nonempty subset of R. A mapping f: R → R is called commuting on S if [f(x),x] = 0 for all x ∈ S. In this paper, firstly, we generalize the well-known result of Posner related to commuting derivations on prime rings. Secondly, we show that if R is a semiprime ring and I is a nonzero ideal of R, then a derivation d of R is commuting on I if one of the following conditions holds: (i) For all x, y ∈ I, either d([x,y]) = [x,y] or d([x,y]) = -[x,y]. (ii) For all x, y ∈ I, either d(x ◦ y) = x ◦ y or d(x ◦ y) = -(x ◦ y). (iii) R is 2-torsion free, and for all x, y ∈ I, either [d(x),d(y)] = d([x,y]) or [d(x),d(y)] = d([y,x]). Furthermore, if d(I) ≠ {0}, then R has a nonzero central ideal. Finally, we introduce the notation of generalized biderivation and prove that every generalized biderivation on a noncommutative prime ring is a biderivation.

On Maps of Period 2 on Prime and Semiprime Rings

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2014/140790 ◽

2014 ◽

Vol 2014 ◽

pp. 1-4

Author(s):

H. E. Bell ◽

M. N. Daif

Keyword(s):

Generalized Derivations ◽

Prime Rings ◽

Period 2 ◽

Semiprime Rings ◽

Rings With Involution

A mapfof the ringRinto itself is of period 2 iff2x=xfor allx∈R; involutions are much studied examples. We present some commutativity results for semiprime and prime rings with involution, and we study the existence of derivations and generalized derivations of period 2 on prime and semiprime rings.

On Prime and Semiprime Rings with Symmetric Generalized Biderivations

Al-Mustansiriyah Journal of Science ◽

10.23851/mjs.v28i1.321 ◽

2017 ◽

Vol 28 (1) ◽

pp. 112

Author(s):

Auday H. Mahmood ◽

Dheaa K. Hussain K. Hussein

Keyword(s):

Semiprime Ring ◽

Prime Rings ◽

The propose of this paper is to present some results concerning the symmetric generalized Biderivations when their traces satisfies some certain conditions on an ideal of prime and semiprime rings. We show that a semiprime ring R must have a nontrivial central ideal if it admits appropriate traces of symmetric generalized Biderivations, under similar hypothesis we prove commutativity in prime rings.

A Note on Differential Identities in Prime and Semiprime Rings

Contemporary Mathematics ◽

10.37256/cm.000127.77-83 ◽

2020 ◽

pp. 77-83

Author(s):

Mohammad Shadab Khan ◽

Mohd Arif Raza ◽

Nadeemur Rehman

Keyword(s):

Prime Ring ◽

Semiprime Ring ◽

Nonzero Ideal ◽

Differential Identities ◽

Standard Identity ◽

Semiprime Rings ◽

Positive Integers

Let R be a prime ring, I a nonzero ideal of R, d a derivation of R and m, n fixed positive integers. (i) If (d ( r ○ s)(r ○ s) + ( r ○ s) d ( r ○ s)n - d ( r ○ s))m for all r, s ϵ I, then R is commutative. (ii) If (d ( r ○ s)( r ○ s) + ( r ○ s) d ( r ○ s)n - d (r ○ s))m ϵ Z(R) for all r, s ϵ I, then R satisfies s4, the standard identity in four variables. Moreover, we also examine the case when R is a semiprime ring.

Derivations of differentially semiprime rings

Asian-European Journal of Mathematics ◽

10.1142/s1793557119500797 ◽

2019 ◽

Vol 12 (05) ◽

pp. 1950079

Author(s):

Ahmad Al Khalaf ◽

Iman Taha ◽

Orest D. Artemovych ◽

Abdullah Aljouiiee

Keyword(s):

Commutative Ring ◽

Semiprime Ring ◽

Prime Rings ◽

Commutative Case ◽

Lie Ring ◽

Lie Rings ◽

Semiprime Rings ◽

Torsion Free

Earlier D. A. Jordan, C. R. Jordan and D. S. Passman have investigated the properties of Lie rings Der [Formula: see text] of derivations in a commutative differentially prime rings [Formula: see text]. We study Lie rings Der [Formula: see text] in the non-commutative case and prove that if [Formula: see text] is a [Formula: see text]-torsion-free [Formula: see text]-semiprime ring, then [Formula: see text] is a semiprime Lie ring or [Formula: see text] is a commutative ring.

Annihilator condition of a pair of derivations in prime and semiprime rings

Indian Journal of Pure and Applied Mathematics ◽

10.1007/s13226-015-0166-z ◽

2016 ◽

Vol 47 (1) ◽

pp. 111-124 ◽

Cited By ~ 2

Author(s):

Basudeb Dhara ◽

Nurcan Argaç ◽

Krishna Gopal Pradhan

Keyword(s):

Generalized Derivations on Power Values of Lie Ideals in Prime and Semiprime Rings

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2014/216039 ◽

2014 ◽

Vol 2014 ◽

pp. 1-8 ◽

Cited By ~ 3

Author(s):

Vincenzo De Filippis ◽

Nadeem UR Rehman ◽

Abu Zaid Ansari

Keyword(s):

Lie Ideal ◽

Generalized Derivations ◽

Fixed Integer ◽

Free Ring ◽

Semiprime Rings ◽

Positive Integers ◽

Torsion Free

LetRbe a 2-torsion free ring and letLbe a noncentral Lie ideal ofR, and letF:R→RandG:R→Rbe two generalized derivations ofR. We will analyse the structure ofRin the following cases: (a)Ris prime andF(um)=G(un)for allu∈Land fixed positive integersm≠n; (b)Ris prime andF((upvq)m)=G((vrus)n)for allu,v∈Land fixed integersm,n,p,q,r,s≥1; (c)Ris semiprime andF((uv)n)=G((vu)n)for allu,v∈[R,R]and fixed integern≥1; and (d)Ris semiprime andF((uv)n)=G((vu)n)for allu,v∈Rand fixed integern≥1.