Let N be a 3-prime 2-torsion-free zero-symmetric left near-ring with multiplicative center Z. We prove that if N admits a nonzero generalized derivation f such that f(N)⊆Z, then N is a commutative ring. We also discuss some related properties.

1. Introduction

Let N be a zero-symmetric left near-ring, not
necessarily with a multiplicative identity element; and let Z be its multiplicative center. Define N to be 3-prime if for all a,b∈N∖{0}, aNb≠{0};
and call N 2-torsion-free if (N,+) has no elements of order 2. A
derivation on N is an additive
endomorphism D of N such that D(xy)=xD(y)+D(x)y for all x,y∈N. A
generalized derivation f with associated derivation D is an additive endomorphism f:N→N such that f(xy)=f(x)y+xD(y) for all x,y∈N.
In the case of rings, generalized derivations have received significant
attention in recent years.

In [1], we proved the following.

Theorem A.

If N is 3-prime and 2-torsion-free
and D is a derivation such that D2=0, then D=0.

Theorem B.

If N is a 3-prime 2-torsion-free
near-ring which admits a nonzero derivation D for which D(N)⊆Z, then N is a
commutative ring.

Theorem C.

If
N is a 3-prime 2-torsion-free
near-ring admitting a nonzero derivation D such that D(x)D(y)=D(y)D(x) for all x,y∈N, then N is a commutative ring.

In this paper, we investigate possible
analogs of these results, where D is replaced by a generalized derivation f.

We will need three easy lemmas.

Lemma 1.1 (see [<xref ref-type="bibr" rid="B1">1</xref>, Lemma 3]).

Let N be a
3-prime near-ring.

Ifz∈Z∖{0}, then z is not a zero divisor.

If Z∖{0} contains an element z such that z+z∈Z, then (N,+) is abelian.

If D is a
nonzero derivation and x∈N is such thatxD(N)={0} or D(N)x={0}, then x=0.

Lemma 1.2 (see [<xref ref-type="bibr" rid="B4">2</xref>, Proposition 1]).

If N is an arbitrary near-ring
and D is a derivation on N, then D(xy)
= D(x)y + xD(y) for all x,y∈N.

Lemma 1.3.

Let N be an arbitrary near-ring and let
f be a generalized derivation on N with associated derivation D. Then

(f(a)b+aD(b))c=f(a)bc+aD(b)c∀a,b,c∈N.

Proof.

Clearly f((ab)c)=f(ab)c+abD(c)=(f(a)b+aD(b))c+abD(c); and by using Lemma 1.2, we obtain f(a(bc))=f(a)bc+aD(bc)=f(a)bc+aD(b)c+abD(c).

Comparing these two expressions for f(abc) gives the desired
conclusion.

2. The Main Theorem

Our best result
is an extension of Theorem B.

Theorem 2.1.

Let N be a
3-prime 2-torsion-free near-ring. If N admits a nonzero generalized derivation f such
that f(N)⊆Z, then N is a commutative ring.

In the proof of this theorem, as well as
in a later proof, we make use of a further lemma.

Lemma 2.2.

Let R be a 3-prime near-ring, and let f be a
generalized derivation with associated derivation D≠0. If D(f(N))={0}, then f(D(N))={0}.

Proof.

We are assuming that D(f(x))=0 for all x∈N. It follows that D(f(xy))=D(f(x)y)+D(xD(y))=0 for all x,y∈N, that is,

f(x)D(y)+D(x)D(y)+xD2(y)=0∀x,y∈N. Applying D again, we get

f(x)D2(y)+D2(x)D(y)+D(x)D2(y)+D(x)D2(y)+xD3(y)=0∀x,y∈N. Taking D(y) instead of y in (2.1) gives f(x)D2(y)+D(x)D2(y)+xD3(y)=0, hence (2.2) yields

D2(x)D(y)+D(x)D2(y)=0∀x,y∈N. Now, substitute D(x) for x in (2.1), obtaining f(D(x))D(y)+D2(x)D(y)+D(x)D2(y)=0; and use (2.3) to conclude that f(D(x))D(y)=0 for all x,y∈N. Thus, by Lemma 1.1(iii), f(D(x))=0 for all x∈N.

Proof of Theorem <xref ref-type="statement" rid="thm2.1">2.1</xref>.

Since f≠0,
there exists x∈N such that0≠f(x)∈Z. Since f(x)+f(x)=f(x+x)∈Z,(N,+) is abelian by Lemma 1.1(ii). To complete the proof, we show that N is multiplicatively commutative.

First, consider the case D =
0, so that f(xy)=f(x)y∈Z for all x,y∈N. Then f(x)yw=wf(x)y,
hence f(x)(yw−wy)=0
for all x,y,w∈N. Choosing x such that f(x)≠0 and invoking Lemma 1.1(i), we get yw − wy = 0 for all y,w∈N.

Now
assume that D≠0, and
let c∈Z∖{0}. Then f(xc)=f(x)c+xD(c)∈Z;
therefore, (f(x)c+xD(c))y=y(f(x)c+xD(c)) for all x,y∈N,
and by Lemma 1.3, we see that f(x)cy+xD(c)y=yf(x)c+yxD(c). Since both f(x) and D(c) are in Z, we have D(c)(xy−yx)=0 for all x,y∈N,
and provided that D(Z)≠{0}, we
can conclude that N is commutative.

Assume now that D≠0 and D(Z)={0}. In particular, D(f(x))=0 for all x∈N. Note that for c∈N such that f(c)=0, f(cx)=cD(x)∈Z;
hence by Lemma 2.2, D(x)D(y)∈Z and D(y)D(x)∈Z for each x,y∈N. If one of these is 0, the other is a central
element squaring to 0, hence is also 0. The remaining possibility is that D(x)D(y) and D(y)D(x) are nonzero central elements, in which
case D(x) is not a zero divisor. Thus D(x)D(x)D(y)=D(x)D(y)D(x) yields D(x)(D(x)D(y)−D(y)D(x))=0=D(x)D(y)−D(y)D(x). Consequently, N is commutative by Theorem C.

3. On Theorems A and C

Theorem C does not extend to generalized
derivations, even if N is a ring. As in [3], consider
the ring H of real quaternions, and define f:H→H by f(x)=ix+xi. It is easy to check that f is a generalized derivation with associated
derivation given by D(x)=xi−ix, and that f(x)f(y)=f(y)f(x) for all x,y∈H.

Theorem A also does not extend to
generalized derivations, as we see by letting N be the ring M2(F) of 2 × 2 matrices over a field F and letting f be defined by f(x)=e12x. However, we do have the following results.

Theorem 3.1.

Let N be a 3-prime near-ring, and let f be a
generalized derivation on N with associated derivation D. If f2=0, thenD3=0. Moreover, if N is 2-torsion-free, then D(Z)={0}.

Proof.

We have
f2(xy)=f(f(x)y+xD(y))=f(x)D(y)+f(x)D(y)+xD2(y)=0∀x,y∈N.
Applying f to (3.1) gives

f(x)D2(y)+f(x)D2(y)+f(x)D2(y)+xD3(y)=0∀x,y∈N. Substituting D(y) for y in (3.1) gives

f(x)D2(y)+f(x)D2(y)+xD3(y)=0; Therefore,
by (3.2) and (3.3),

f(x)D2(y)=0∀x,y∈N. It
now follows from (3.3) that xD3(y)=0 for all x,y∈N;
and since N is 3-prime, D3=0.

Suppose now that N is 2-torsion-free and that D(Z)≠{0},
and let z∈Z be such that D(z)≠0. Then if x,y∈N and f(N)x={0}, then f(yz)x=f(y)zx+yD(z)x=0=yD(z)x; and since N is 3-prime and D(z) is not a zero divisor, x = 0. It now follows from (3.4)
that D2=0 and hence by Theorem A that D = 0. But this contradicts our assumption that D(Z)≠{0},
hence D(Z)={0} as claimed.

Theorem 3.2.

Let N be a 3-prime and 2-torsion-free
near-ring with 1. If f is a
generalized derivation on N such that f2=0 and f(1)∈Z, then f=0.

Proof.

Note that f(x)=f(1x)=f(1)x+1D(x), so

f(x)=cx+D(x),c∈Z. If c =
0, then f=D and D2=0, so D =
0 by Theorem A and therefore f =
0.

If c≠0,
then c is not a zero divisor, hence by (3.4) D2=0 and D = 0. But then f(x)=cx and f2(x)=c2x=0 for all x∈N. Since c2 is not a zero divisor, we get N={0}—a contradiction. Thus, c = 0 and we are finished.

4. More on Theorem C

In [4], the author studied generalized
derivations f with associated
derivation D which have the additional
property that

f(xy)=D(x)y+xf(y)∀x,y∈N. Our
final theorem, a weak generalization of Theorem C, was stated in [4]; but the
proof given was not correct. (At one
point, both left and right distributivity were assumed.) We now have all the results required for a
proof.

Theorem 4.1.

Let N be a 3-prime 2-torsion-free near-ring which
admits a generalized derivation f with nonzero associated derivation D such
that f satisfies (∗). If f(x)f(y)=f(y)f(x) for all x,y∈N, then N is a commutative ring.

Proof.

It is correctly shown in [4] that
(N,+) is abelian and either f(N)⊆Z or D(f(N))={0}. Hence, in view of Theorem 2.1, we may assume
that D(f(N))=0 and therefore, by Lemma 2.2,
that f(D(N))={0}. We calculate f(D(x)D(y)) in two ways. Using the defining property of f, we obtain f(D(x)D(y))=f(D(x))D(y)+D(x)D2(y)=D(x)D2(y); and using (∗), we obtain f(D(x)D(y))=D2(x)D(y)+D(x)f(D(y))=D2(x)D(y). Thus, D2(x)D(y)=D(x)D2(y) for all x,y∈N. But since D(f(N))={0}, (2.3) holds
in this case as well; therefore D2(x)D(y)=0 for all x,y∈N,
hence by Lemma 1.1(iii) D2=0. Thus, D = 0, contrary to our original
hypothesis, so that the case D(f(N))={0} does not in fact occur.

Acknowledgment

This research is supported by the Natural
Sciences and Engineering Research Council of Canada, Grant no. 3961.

BellH. E.MasonG.BetschG.On derivations in near-ringsWangX.-K.Derivations in prime near-ringsBellH. E.RehmanN.-U.Generalized derivations with commutativity and anti-commutativity conditionsGölbaşiÖ.Notes on prime near-rings with generalized derivation