# On Multiplicative (Generalized)-Derivation Involving Semiprime Ideals

Authors: Hafedh M. Alnoghashi; Sabah Naji; Nadeem ur Rehman
Journal: Journal of Mathematics
 (2023)

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## Abstract

LetA be any arbitrary associative ring, P a semiprime ideal, and J a nonzero ideal of A. In this study, using multiplicative (generalized)-derivations, we explore the behavior of semiprime ideals that satisfy certain algebraic identities. Moreover, examples are provided to demonstrate that the restrictions imposed on the hypotheses of the various theorems are necessary.


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## Body

## 1. Introduction

Throughout this study,A represents an associative ring with the center ZA. The symbols r°t and r,t, where r,t∈A, stand for the anticommutator rt+tr and commutator rt−tr, respectively. We will frequently use the basic commutator and anticommutator identities: irt,s=rt,s+r,st, r,tt=tr,s+r,ts and iir°tt=r°ts−tr,s=tr°s+r,ts, rt°s=rt°s−r,st=r°st+rt,s for all r,t,s∈A. A ring A is said to be a prime ring if for all r,s∈A, rrt=0 implies r=0 or s=0, and A is said to be a semiprime ring if for all r∈A, rrr=0 implies r=0. An ideal P is said to be a prime ideal of A if A≠P and for all r,t∈A, rrt⊆P implies r∈P or t∈P, and P is a semiprime ideal if A≠P and for any r∈A, rrr⊆P implies r∈P. A mapping f:D⟶A is said to be a S-commuting mapping on D if fr,r∈S for all r∈D where S,D⊆A. In particular, if S=0, then f is said to be a commuting mapping on D if fr,r=0. Note that every commuting mapping is a S-commuting mapping (put 0=S), but the converse can be inaccurate in general (taking S as a set of A which has no zero such that fr,r∈S, then f is a S-commuting mapping, but it is not a commuting mapping). An additive mapping μ:A⟶A is said to be a derivation of A if μrt=μrt+rμt holds for all r,t∈A. An additive mapping G:A⟶A associated with a derivation μ:A⟶A is said to be a generalized derivation of A if Grt=Grt+rμt holds for all r,t∈A.Daif [1] introduced the concept of multiplicative derivations, and it was motivated by the work of Martindale [2]. Dhara and Ali [3] have expanded the concept of multiplicative derivation to include multiplicative (generalized)-derivation. As a result, a mapping G:A⟶A (not necessarily additive) is called multiplicative (generalized)-derivation if Grt=Grs+rψs holds for all r,s∈A, where ψ is any mapping (not necessarily a derivation nor an additive map). As a result, multiplicative (generalized)-derivation encompasses both multiplicative and multiplicative generalized derivation ideas. The ideas of multiplicative centralizers are covered by a multiplicative (generalized)-derivation associated with mapping ψ=0 (not necessarily additive). However, there are few articles on this subject (see [3–5] for a partial bibliography).Every generalized derivation is obviously a multiplicative (generalized)-derivation onA. However, the converse can be inaccurate in general. The following example shows this fact.Example 1.
LetD be any ring and let(1)A=0rt00s000:r,t,s∈D.
We define the mapsμ and G:A⟶A as follows: (2)μ0rt00s000=00r2000000,G0rt00s000=00ts000000.
Then, it is simple to show thatG is a multiplicative (generalized)-derivation associated with a multiplicative derivation μ, but G is not a generalized derivation of A.
Numerous studies have revealed that the global structure ofA is typically related to the behavior of additive mappings formed on A. Recent work has investigated the commutativity of the factor ring A/P, where P is the prime ideal of any arbitrary ring A, using algebraic identities in P, including derivations and generalized derivations (see [6–11]).
In 2021, Mamouni et al. [12] proved that any ring A/P must be commutative or GA⊆P, where P is a prime ideal of A if A admits a generalized derivation G satisfying Grt¯∈ZA/P for all r,s∈A. In the same year, Rehman et al. [13] proved that any ring A/P must be commutative or μA,φA⊆P, where P is a prime ideal of A if A admit generalized derivations G,μ and ℋ,φ satisfying Gr,ℋs±r,s∈P for all r,s∈A.
We investigate algebraic identities with multiplicative (generalized)-derivation involving semiprime ideal in this article without making any assumptions about semiprimeness on the ring in discussion. In addition, examples are given to show the need of the constraints imposed on the various theorems’ hypotheses.


## 2. The Main Results

Lemma 1.
LetA be a ring with P a semiprime ideal and J a nonzero ideal of A. If s∈J such that tst⊆P, then s∈P. In particular, if s∈J such that ts⊆P or st⊆P, then s∈P.Proof.
Assume thats∈J such that tst⊆P. That is, ttt∈P for all t∈J. Replacing t by atb in the last relation, where a,b∈A, we have tatbt∈P for all a,b∈A. That is, trtrt⊆P. By semiprimeness of P, we obtain s∈P, as desired.
Now, ifs∈J such that ts⊆P or st⊆P, then tst⊆P, and so, similar to the earlier, we get s∈P, as desired.Theorem 1.
LetA be a ring with P a semiprime ideal and J a nonzero ideal of A and ψ any mapping on A. If G,μ and ℋ,φ are multiplicative (generalized)-derivations of A which satisfy(3)ℋrt+GrGt+ψr,t∈P,for all r,t∈J, then μ and φ are P-commuting on J. Moreover, if ψ is an automorphism, then A/P is commutative.Proof.
Assume that(4)ℋrt+GrGt+ψr,t∈Pfor allr,t∈J.
Replacingt by tt in (4) and using it, where s∈J, we have(5)rtφs+Grtμs+tψr,s∈P.
Left multiplying (5) by a, where a∈A, we get(6)rrtφs+aGrtμs+atψr,s∈P.
Takingt by at in (5), we refer that(7)ratφs+Gratμs+atψr,s∈P.
Comparing (6) and (7) gives(8)r,atφs+Gr,atμs∈P.
Puttingt by at in (8), we obtain(9)r,aatφs+Gr,aatμs∈P.
Again, puttingr by rr in (8), we see that(10)r,aatφs+Gra,atμs∈P.
That is,(11)r,aatφs+Gra+rμa,atμs∈P.
Hence,(12)r,aatφs+Gr,aatμs+rμa,atμs∈P.
Comparing (9) and (12), we find that rμa,atμs∈P. Putting t by tr in the last relation, we have(13)rμa,atrμs∈P.
Right multiplying the previous expression bys and then replacing t by tt, we get(14)rμa,atrμs,s∈P.
Takinga by s in the previous expression, we see that rμs,strμs,s∈P. Using Lemma 1 in the last relation, we get rμs,s∈P. Taking r by ar in the last relation and using it, where a∈A, we get a,srμs∈P. It is clear to see that a,srμs,s∈P. In particular, μs,srμs,s∈P. By using Lemma 1 in the last relation, we obtain(15)μs,s∈Pfor alls∈sas desired.
Now, right multiplying (8) by s and then replacing t by tt and then subtracting them, we get(16)r,atφs,s+Gr,atμs,s∈P.
Using (15) in the last relation, we r,atφs,s∈P. Taking a=φs in the last relation and using Lemma 1, we see that(17)φs,s∈Pfor alls∈sas desired.
Now, assume thatψ is an automorphism. By using (15) and (17) in (7), we obtain atψr,s,s∈P. Since ψ is an automorphism, we get atr,s,s∈P. Since J is a semiprime ideal, we see that(18)tr,s,s∈P.
That is,(19)r,s,str,s,s∈P.
Hence,(20)r,s,sJr,s,s⊆P.
Using Lemma1 in the previous expression, we infer that(21)r,s,s∈Pfor allr,s∈J.
Replacings by s+t in (21) and using it, where t∈J, we obtain r,s,t+r,t,s∈P. Taking t by tt in the previous expression and using it, we have r,s,st+r,st,s∈P. It follows that 2r,s,st+r,st,s∈P. Using (21) in the last relation, we get r,st,s∈P. Putting t by tr in the previous expression and using it, we see that r,str,s∈P. Using Lemma 1 in the previous expression, we conclude that r,s∈P. Replacing r by ra in the previous expression and using it, where a∈A, we obtain ra,s∈P. Again, using Lemma 1 in the last relation, we have a,s∈P. Similarly, from the last relation, we can get a,b∈P for all a,b∈A. It follows that A/P is commutative.Corollary 1.
LetA be a ring with P a semiprime ideal and J a nonzero ideal of A. If G,μ and ℋ,φ are multiplicative (generalized)-derivations of A satisfying(22)ℋrt+GrGt+tr∈Pfor all r,t∈J, then A/P is commutative.Proof.
Sinceℋ is a multiplicative (generalized)-derivation, we get ℋ+1 (where 1 is an identity map) is a multiplicative (generalized)-derivation. Now, put ℋ=ℋ+1 and ψ=−1 in Theorem 1.Theorem 2.
LetA be a ring with P a semiprime ideal and J a nonzero ideal of A. If G,μ and ℋ,φ are multiplicative (generalized)-derivations of A satisfying any one of the conditions(1)
ℋrt+GrGt+r,t∈P(2)
ℋrt+GrGt+r°t∈Pfor all r,t∈J, then A/P is commutative.Proof.
(1)
Puttingψ=1 in Theorem 1, we get the desired result.(2)
Assume that(23)ℋrt+GrGt+r°t∈Pforallr,t∈J.
Takingt by tt in the previous expression and using it, where s∈J, we have(24)rtφs+Grtμs−tr,s∈P.
Now, sinceψ is any mapping in Theorem 1, we can put ψ=−1 in (5), and then using the same technique, we get (15) and (17), and by using both equations in (24), we obtain (18), and so we get the desired result.Corollary 2.
LetA be a ring with P a semiprime ideal and J a nonzero ideal of A. If G,μ and ℋ,φ are multiplicative (generalized)-derivations of A satisfying any one of the conditions(1)
ℋrt+GrGt∈P(2)
ℋrt+GrGt+rt∈Pfor all r,t∈J, then μ and φ are P-commuting on J.Proof.
(1)
Sinceψ is any mapping in Theorem 1, we can put ψ=0, and so we get the desired result.(2)
Puttingℋ by ℋ−1 in (5), we obtain (4).Theorem 3.
LetA be a ring with P a semiprime ideal and J a nonzero ideal of A and ψ any mapping on A. If G,μ and ℋ,φ are multiplicative (generalized)-derivations of A satisfying(25)ℋrt+GtGr+r,ψt∈Pfor allr,t∈J,then μ is a P-commuting on J.Proof.
Assume that(26)ℋrt+GtGr+r,ψt∈Pfor allr,t∈J.
Replacingr by rt in (26), where s∈J, we have(27)ℋrt+GtGr+r,ψts+ℋrtt−ℋrts+Gtrμs+rs,ψt∈P.
By using (26) in (27), we get(28)ℋrs,t+rφst+rtφt−rφts+Gtrμs+rs,ψt∈P.
Puttingr by tr in (28) and then left multiplying it by s and then subtracting them, we see that(29)ℋtr−tℋrs,t+Gt,srμs∈P.
Takingt by tt in (29) and then right multiplying it by s and then subtracting them, we find that(30)Gtt,srμs−Gt,srμss∈P.
That is,(31)Gts+tμs,srμs−Gt,srμss∈P.
Hence,(32)Gt,strμs+tμs,srμs−Gt,srμss∈P.
It follows that(33)Gt,ss,rμs+tμs,srμs∈P.
Replacingt by tt in the last relation and then r by tr in it and then subtracting them, we infer that(34)tμs,ss,rμs+ttμs,srμs−tμs,strμs∈P.
That is,(35)tsμstrμs−μstrμss+ttμsrμss−sμsrμs∈P.
Puttingt by at in the last relation and then left multiplying it by a and then subtracting them, where a∈A, we have a,stμsrμss−sμsrμs∈P. Hence, a,stμsrμs,s∈P. Taking a by μsrμs in the last relation, we get μsrμs,stμsrμs,s∈P. Using Lemma 1 in the last relation, we get(36)μsrμs,s∈P.
Takingr by rμst in (36), where t∈J, we see that μsrμstμs,s∈P. Using (36) in the last relation, we find that μsrμstμs,s∈P. Putting r by tμs,sr in the last relation, we get(37)μstμs,srμstμs,s∈P.
Using Lemma1 in the last relation, we find that μstμs,s∈P. Using the last relation in (36), we get μs,srμs∈P. It follows that μs,srμs,s∈P. Using Lemma 1 in the last relation, we see that μs,s∈P for all s∈J.
Whenψ=0 and ℋ=G in Theorems 1 and 3, we have the following corollary.Corollary 3.
LetA be a ring with P a semiprime ideal and J a nonzero ideal of A. If G,μ is a multiplicative (generalized)-derivation of A satisfying any one of the conditions(1)
Grt+GrGt∈P(2)
Grt+GtGr∈Pfor all r,t∈J, then μ is a P-commuting on J.Theorem 4.
LetA be a ring with P a semiprime ideal and J a nonzero ideal of A. If G,μ is a multiplicative (generalized)-derivation of A satisfying Gr°t+μr°t∈P for all r,t∈J, then μ is a P-commuting on J.Proof.
(1)
Assume that(38)Gr°t+μr°t∈Pfor allr,t∈J.
Puttingt by tr in (38), we have(39)Gr°t+μr°tr+rtμr+2trμr−tμrr∈P,and by using (39) in the last relation, we get(40)rtμr+2trμr−tμrr∈P.
Replacingt by μrt in (40), we obtain(41)rμrtμr+2μrtrμr−μrtμrr∈P.
Left multiplying byμr in (40) and then subtracting from (41), we see that(42)r,μrtμr∈P.
Again, replacingt by tr in the last relation and then right it by r and then subtracting them, we infer that(43)r,μrtr,μr∈P.
Using Lemma1 in the last relation, we get r,μr∈P for all r∈J.Theorem 5.
LetA be a ring with P a semiprime ideal and J a nonzero ideal of A. If G,μ is a multiplicative (generalized)-derivation of A satisfying Gr,t+μr,t∈P for all r,t∈J, then μ is a P-commuting on J.Proof.
Puttingt by r in our hypothesis, we get μ is a P-commuting on J.Theorem 6.
LetA be a ring with P a semiprime ideal and J a nonzero ideal of A. If G,μ and ℋ,φ are multiplicative (generalized)-derivations of A satisfying any one of the conditions(1)
ℋrt+Gr,t+tr∈P(2)
ℋrt+Gr°t+tr∈P(3)
ℋrt+Gr,t+r,t∈P(4)
ℋrt+Gr,t+r°t∈Pfor all r,t∈J, then J,φJ⊆P (moreover, φrr∈P for all r∈J), and μ and G are P-commuting on J.Proof.
(1)
Assume that(44)ℋrt+Gr,t+tr∈Pfor allr,t∈J.Takingt by tt in (44), where s∈J, we have(45)ℋrts+rtφs+Gr,ts+tGr,s+ttr∈P.By using (44) in the last relation, we get(46)rtφs+tGr,s+ts,r∈P.Puttingt=bt in (46) and then left multiplying it b and then subtracting them, where b∈A, we obtain r,btφs∈P. Thus, r,btr,φs∈P. Taking b=φs in the last relation, we conclude r,φstr,φs∈P. Using Lemma 1 in the last relation, we get r,φs∈P. That is,(47)J,φJ⊆Pas desired. Takingr by ra in (47) and using it, where a∈A, we get ra,φs∈P. That is,(48)JA,φs⊆Pfor alls∈J.From (46), we get rtφs+tGr−r,s∈P. Putting t by ta in the previous expression and then right multiplying it by a and then subtracting them, where a∈A, we obtain(49)rtφs,a+tGr−r,s,a∈P.Using (48) in the last relation, we see that tGr−r,s,a∈P. Taking t by Gr−r,s,ab in the last relation, where b∈A, we find that Gr−r,s,abGr−r,s,a∈P. That is, Gr−r,s,a∈P. Replacing s by ta in the previous expression and using it, we infer that sGr−r,a,a∈P. Again, replacing s by bt in the previous expression and using it, where b∈A, we conclude that b,asGr−r,a∈P. Putting b=Gr−r and a=t in the previous expression, where t∈J, we have Gr−r,tsGr−r,t∈P. Using Lemma 1 in the previous expression, we obtain(50)Gr−r,t∈P.In particular,Gr,r∈P for all r∈J, as desired. Putting r=rt in (50) and using it, we get rμt,t∈P. Replacing r by μtr in the previous expression and using it, we see that μt,trμt∈P. It follows that μt,trμt,t∈P. Using Lemma 1 in the previous expression, we infer that μt,t∈P for all t∈J, as desired.(2)
Assume that(51)ℋrt+Gr°t+tr∈Pfor allr,t∈J.Replacingtt instead of t in (51), where s∈J, we have(52)ℋrts+rtφs+Gr°ts−tGr,s+trt+ts,r∈P.By using (51) in (52), we get(53)rtφs−tGr,s+ts,r∈P.Now, similar to (46), we conclude the desired result.
From (6) and (7), since ℋ is a multiplicative (generalized)-derivation, we infer that ℋ±1 (where 1 is an identity map) is a multiplicative (generalized)-derivation; now putting ℋ=ℋ±1 in (44), we get the desired result.Corollary 4.
LetA be a ring with P a semiprime ideal and J a nonzero ideal of A. If G,μ and ℋ,φ are multiplicative (generalized)-derivations of A satisfying any one of the conditions(1)
ℋrt+Gr,t∈P(2)
ℋrt+Gr,t+rt∈P(3)
ℋrt+Gr°t∈P(4)
ℋrt+Gr°t+rt∈P(5)
ℋrt+Gr°t+r°t∈P(6)
ℋrt+Gr°t+r,t∈Pfor all r,t∈J, then J,φJ⊆P (moreover, φrr∈P for all r∈J), and μ and G are P-commuting on J.Proof.
TakingG by G−1 and ℋ by ℋ+1 in (1) and then using Theorem 6 (1), we get (1). Taking ℋ by ℋ−1 in (2), we get (1), and then using the previous trick, we obtain (2). Taking ℋ by ℋ−1 and G by G+1 in (3) and then using Theorem 6 (2), we get (3). Taking ℋ by ℋ−1 in (4) and then using (3), we get (4). Taking G by G−1 in (5) and then using (3), we get (5). Taking ℋ by ℋ−1 and G by G+1 in (6) and then using (4), we get (6)..Theorem 7.
LetA be a ring with P a semiprime ideal and J a nonzero ideal of A. If G,μ and ℋ,φ are multiplicative (generalized)-derivations of A satisfying any one of the conditions(1)
ℋrt+r,Gt+tr∈P(2)
ℋrt+r,Gt+r,t∈P(3)
ℋrt+r,Gt+r°t∈P(4)
ℋrt+r,Gt+rt∈P(5)
ℋrt+r,Gt∈Pfor all r,t∈J, then μr,rr∈P and φrr∈P for all r∈J.Proof.
(1)
Assume that(54)ℋrt+r,Gt+tr∈Pfor allr,t∈J.
Replacingt by tr in (54) and using it, we have(55)rtφr+r,tμr+tr,μr∈P.
Puttingbt by t in (55) and then left multiplying it by b, where b∈A, and then subtracting them, we get(56)r,btφr+r,btμr∈P.
Takingt by rt in (55) and then left multiplying it by r,b and using (56), we see that(57)r,btμrr∈P.
Again, takingb=μrr in (57), this gives(58)r,μrrtμrr∈Pand so r,μrrtr,μrr∈P. Using Lemma 1 in the last relation, we get(59)r,μrr∈Pfor allr∈s,as desired.
Puttingt by μrrar,b in (57), where a∈A, we get(60)r,bμrrar,bμrr∈Pand so r,bμrr∈P. In particular, r,tμrr∈P. Now, right multiplying (55) by r and using the last relation and (59), we get rtφrr∈P. Left multiplying the last relation by φr, we obtain φrrtφrr∈P. Using Lemma 1 in the previous expression, we see that φrr∈P for all r∈J, as desired.
(5)–(8) are similar to (55).Theorem 8.
LetA be a ring with P a semiprime ideal and J a nonzero ideal of A. If G,μ and ℋ,φ are multiplicative (generalized)-derivations of A satisfying(61)ℋrt+r°Gt+tr∈Pfor all r,t∈J, then μr,rr∈P and φr,rr∈P for all r∈J.Proof.
Assume that(62)ℋrt+r°Gt+tr∈Pfor allr,t∈J.
Replacingt by tr in (62), we have(63)ℋrtr+rtφr+r°Gtr+rtμr+tμrr+tr2∈P.
By using (62) in the last relation, we get(64)rtφr+rtμr+tμrr∈P.
Now, similar to the proof in Theorem7(1), equation (55), we get μr,rr∈P and φr,rr∈P for all r∈J.
From Theorem8, we get the next result.Corollary 5.
LetA be a ring with P a semiprime ideal and J a nonzero ideal of A. If G,μ and ℋ,φ are multiplicative (generalized)-derivations of A satisfying any one of the conditions(1)
ℋrt+r°Gt+r,t∈P(2)
ℋrt+r°Gt+r°t∈P(3)
ℋrt+r°Gt+rt∈P(4)
ℋrt+r°Gt∈Pfor all r,t∈J, then μr,rr∈P and φr,rr∈P for all r∈J.Example 2.
LetA=rt0s:r,t,s∈ℤ and ψ is an identity mapping on A. We define G=μ=ℋ=φ:A⟶A as(65)Grt0s=0t00.
It is easy to verify thatG and ℋ are multiplicative (generalized)-derivations on A associated with the mapping μ and φ on A, respectively. Let(66)J=0t00:t∈ℤand P=0000. Here, we see that J is an ideal of A satisfying the following conditions:(1)
ℋrt+GrGt+r,t∈P(2)
ℋrt+GrGt+r°t∈P(3)
ℋrt+GrGt+tr∈Pfor all r,t∈J but A/P is noncommutative. Note that 0100A0100⊆P but 0100⊆P, and so P is not a semiprime ideal. Hence, the semiprimeness of our hypothesis is essential in Theorem 2 and Corollary 4.



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