A Work on Generalized Reverse Derivation and Skew-Derivation on Prime Near-Rings

A Work on Generalized Reverse Derivation and Skew-Derivation on Prime Near-Rings

Abdu Madugu and Tasiu Abdullahi Yusuf^*

Department of Mathematics and Statistics, Faculty of Natural and Applied Sciences, Umaru Musa Yar’adua University, Katsina, Nigeria

*Corresponding author

DOI: https://dx.doi.org/10.47772/IJRISS.2025.90400374

Received: 20 March 2025; Accepted: 29 March 2025; Published: 17 May 2025

ABSTRACT

This paper investigates the analysis of prime near-ring by explore the detailed structure of the generalized derivations that satisfy some specific assumptions. Let N be a prime near-rings and G be a generalized reverse derivative associated with mapping d on N. An additive mapping d:N→N is said to be a derivation on N if d(xy)=d(x)y+xd(y) for all x,y ∈N. A mapping G: N→N associated with derivation d is called a generalized derivation on N if G(xy)=G(x)y+xd(y) for all x,y∈N. Also, a mapping d: N→N is said to be a reverse derivation on N if d(xy)=d(y)x+yd(x) for all x,y ∈N and a mapping G:N→N associated with reverse derivation d is said to be a generalized reverse derivation on N if G(xy)=G(y)x+yd(x) for all x,y∈N. We prove some results on commutativity of prime near-rings involving generalized reverse derivations. In addition, we prove that; for prime near-rings N, if d(x)d(y)±xy=0 for all x,y∈N then d=0 where d is a skew- derivation associated with an automorphism β∶ N→N. Furthermore, for a prime near-ring N with generalized derivative G associated with mapping d on N, if G(x)G(y)±xy=0 for all x,y∈G then d=0.

Keywords: Prime near-ring, reverse derivation, generalized reverse derivation, skew derivation.

INTRODUCTION

In this context, the symbol [x,y] and (xoy), represent the Lie product xy-yx and Jordan product xy+yx respectively, where x,y∈N. A near-ring N is called prime near-rings if aNb={0} for anya, b∈N, implies that either a=0 or b=0 and is said to be semiprime near-rings if aNa={0} for any a∈N, thena=0.

Bresar [1] defined an additive mapping according to him a mapping f is said to be an additive mapping on R if f(x+y)=f(x)+f(y) for all x,y∈R. According to [2] a mapping d:R→R is said to be a derivation if d(xy)=d(x)y+xd(y), for all x,y∈R. If d is an additive mapping then d is said to be a derivation on R. Also an additive mapping F:R→R is called generalized derivation if there exist a derivation d:R→R such that F(xy)=F(x)y+xd(y), for all x,y∈R.

The notion of multiplicative derivation was first introduced by Daif [3] according to him a mapping D:R→R is called multiplicative derivation if it satisfies D(xy)=D(x)y+xD(y), for all x,y∈R where in this, the mappings are not supposed to be an additive. Further Daif and Tammam El-sayiad [4] extended multiplicative derivation to multiplicative generalized derivation that is a mapping F on R is said to be a multiplicative generalized derivation if there exist a derivation d on R such that F(xy)=F(x)y+xd(y), for all x,y∈R. From the definition of multiplicative generalized derivation if d is any mapping not necessarily additive and derivation then F is said to be multiplicative (generalized) derivation. Recently Dhara and Ali [5] give a more precise definition of multiplicative (generalized)derivation as follows: A mapping F:R→R is said to be a multiplicative (generalized) derivation if there exist a map g on R such that F(xy)=F(x)y+xg(y), for all x,y∈R. Where g is any mapping on R (not necessarily additive). Therefore, the concept of multiplicative (generalized) derivation covers the concept of multiplicative derivation and multiplicative generalized derivation.

The notion of reverse derivation was first introduce by Herstein [6] According to him an additive mapping d on R is said to be reverse derivation if d(xy)=d(y)x+yd(x), for all x,y∈R. While, according to [7] the generalized reverse derivation is an additive mapping F:R→R if there exist a map d:R→R such that F(xy)=F(y)x+yd(x), for all x,y∈R. A map F:R→R is called multiplicative (generalized)-reverse derivation if F(xy)=F(y)x+yd(x), for all x,y∈R, where d is any map on R and F is not necessarily additive [8].

Motivated by the above concepts, we prove the commutativity of prime near-rings involving generalized reverse derivations on a prime near-rings N and result on skew-derivation of the same N.

Results on Prime near-rings N.

Theorem 1.1

Let N be prime near-ring and G be a generalized derivative associated with mapping don N. If G(x)G(y)±xy=0 for all x,y∈G then d=0.

Proof.

First we consider the case

G(x)G(y)+xy=0             (1)

for all x,y,∈N. Substituting yz instead of y in equation (1), we obtain

G(x)G(yz)+xyz=0

G(x)(G(y)z+yd(z))+xyz=0

G(x)G(y)z+G(x)yd(z)+xyz=0

But G(x)G(y)=-xy

Therefore, -xyz+G(x) yd(z)+xyz=0

G(x)yd(z)+xyz-xyz=0, for all x,y,z∈N         (2)

Substituting xr instead of x in equation (2), we get

G(xr)yd(z)=0

((G(x)r+xd(r))yd(z)=0

G(x)ryd(z)+xd(r))yd(z)=0 ∀ x,y,z∈N         (3)

Substituting ry instead of y in equation (2), we obtain

G(x)ryd(z)=0         (4)

Subtracting equation (3) from equation (4), we obtain

xd(r)yd(z)=∀ x,y,r,z∈N         (5)

Replacing xd(r) by d(t) in (5), we get

d(t)yd(z)=0 ∀ y,t,z∈N

Since d is mapping on N ∀ y,t,z∈N.

This implies that, d(t)Nd(z)=0 ∀ t,z∈N

Therefore, by primeness of N, we obtain d(t)=0 or d(z)=0.

Using similar approach, we can prove that the same result holds for

G(x)G(y)-xy=0 ∀ x,y,r,z∈G.

RESULTS ON COMMUTATIVITY OF N.

Theorem 1.2

Let N be prime near-ring and G be a generalized reverse derivative associated with mapping don N. If G(x)G(y)±xy=0 for all x,y∈N then N is commutative.

Proof:

First we consider the case,

G(x)G(y)+xy=0         (6)

for all x,y,∈N. Substituting zy instead of y in equation (1), we obtain

G(x)G(zy)+xy=0, for all x,y,z∈N.

By definition of generalized derivation, we have

G(x)(G(y)z+yd(z))+xy=0

G(x)(G(y)z+G(x)yd(z))+xy=0

But G(x)G(y)=-xy

-xyz+G(x)yd(z)+xzy=0

xzy-xyz+G(x)yd(z)=0

-x([z,y])+G(x)yd(z)=0

x[z,y]+G(x)yd(z)=0

G(x)yd(z)+x[z,y]=0         (7)

for all x,y,∈N. Substituting ry instead of y in equation (7) where r∈I, we obtain

G(x)ryd(z)+x[z,ry]=0 ∀ x,y,r,z∈N

G(x)ryd(z)+x(r[z,y]+[z,r]y)=0

G(x)ryd(z)+xr[z,y]+x[z,r]y=0         (8)

Replacing x instead of with rx in equation (7), we get

G(rx)yd(z)+rx[z,y]=0        (9)

By definition of generalized reverse derivation, we get

(G(x)r+xd(r))+rx[z,y]=0

G(x)ryd(z)+xd(r)yd(z)+rx[z,y]=0        (10)

Subtracting equation (8) by (10), we get

xr[z,y]+x[z,r]y-xd(r)yd(z)-rx[z,y]=0

x[z,r]y-xd(r)yd(z)+[x,r][z,y]=0         (11)

for all x,y,r,z∈N.

Substituting tx instead of x in equation (11), we obtain

tx[z,r]y-txd(r)yd(z)+[tx,r][z,y]=0

tx[z,r]y-txd(r)yd(z)+(t[x,r]+[t,r]x)[z,y]=0

tx[z,r]y-txd(r)yd(z)+t[x,r][z,y]+[t,r]x[z,y]=0         (12)

Multiplying equation (11) by t on the left side, we obtain

tx[z,r]y-txd(r)yd(z)+t[x,r][z,y]=0        (13)

Subtracting equation (11) by (12), we get

[x,r]x[z,y]=0 ∀ x,y,r,z∈R

[t,r]N[z,y]=0 ∀ t,y,r,z∈R

By primeness of N, then either [t,r]=0 or [z,y]=0

that is,        tr-rt=0 or zy-yz=0.

Therefore, tr=rt or zy=yz which implies that N is commutative.

Results on Skew Derivation of N.

Theorem 1.3

Let N be prime near-ring and d be a skew- derivative associated with an automorphism

β:N→N.

If d(x)d(y)±xy=0 for all x,y∈N then d=0.

Proof:

First, we consider the case

d(x)d(y)+xy=0        (14)

for all x,y,∈N. Substituting yz instead of y in equation (14), we obtain

d(x)d(yz)+xyz=0

By definition of skew derivation, we get

d(x)(d(y)z+β(y)d(z))+xyz=0 ∀ x,y,z ∈N

d(x)d(y)z+d(x)β(y)d(z)+xyz=0

But d(x)d(y)=-xy

-xyz+d(x)β(y)d(z+xyz)=0

d(x)(β(y)d(z))+xyz-xyz=0

d(x)(β(y)d(z))=0         (15)

Replacing xr instead x in equation (15), we obtain

d(xr)(β(y)d(z))=0

By definition of skew derivation, we have

(d(x)r(β(x)d(r))β(y)d(z)=0

d(x)rβ(y)d(z)+β(x)d(r)β(y)d(z)=0         (16)

Replacing rβ(y) instead of β(y) in equation (15), we get

d(x)rβ(y)d(z)=0 ∀ x.y,r,z∈N        (17)

Subtracting equation (17) equation (16), we obtain

β(x)d(r)β(y)d(z)=0 ∀ x.y,r,z∈N        (18)

Replacing d(r) instead of β(x)d(r) in equation (18), we get

d(r)β(y)d(z)=0 ∀ y,r,z∈N

Since β is an automorphism of N and d is a skew derivation of N and x.y,r,z∈N

This implies that,

d(r)Nd(z) ∀ r,z∈N

By primeness of N, this implies that

d(r)=0 or d(z)=0.

Hence, we obtained the require result.

CONCLUSION

In this paper, we prove the commutativity of prime near-rings with generalized reverse-derivations and investigate whether prime near-rings that admit a nonzero multiplicative reverse-derivation satisfying certain algebraic (or differential) identities are commutative ring. In addition, we show that in a prime near-rings N with a skew-derivative d associated with an automorphism β:N→N if the differential identity d(x)d(y)±xy=0 satisfy for all x,y∈N then the skew-derivation d is zero.

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