WHEN IS A MULTIPLICATIVE DERIVATION ADDITIVE?

Internat. J. Math. & Math. Sci.

VOL. 14 NO. 3 (1991) 615-618

6]5

WHEN IS A MULTIPLICATIVE DERIVATION ADDITIVE?

MOHAMAD

NAGY

DAIF Department

of Mathematics

Faculty of Education IrL Ai-Qra University

Tail, Saudi Arabia

(Received March 29, 1990 and in revised form December 19, 1990)

ABSTRACT.

Our main objective in this note is to prove the following.

Suppose R

is a ring having an idempotent element e

(eO,

el) which satisfies:

(I

I)

^{xR=O implies x=O.}

(M

2)

^{eRx=O implies x=O (and hence Rx=O implies}

(M3)

exeR(l-e)=O implies exe=O.

If d is any multiplicative derivation of

R,

then d is additive.

KEY

WORDS

AND PHRASES.

Ring, idempotent element, derivation, Peirce decomposition.

1980 AMS

SUBJECT CLASSIFICATION CODES.

16A15, 16A70.

I.

INTRODUCTION.

In [I],

Martindale has asked the following question When is a multiplicative mapping additive ? He answered his question for a multiplicative isomorphism of a ring R under the existence of a family of idempotent elements in

R

which satisfies some conditions.

Over the past few years, many results concerning derivations of rings have been obtained.

In

this note, we introduce the definition of ^a multiplicative derivation of a ring

R

to be a mapping d of

R

into

R

such that

d(a) d(a)b + ad(b),

for all a,b in

R.

As Martindale did, we raise the following question

EDen

is a multipl- icative derivation additive? Fortunately, we can give a full answer for this question using Martindale’s conditions when assumed for a single fixed idempotent in

R.

In

the ring

R,

let e be an idempotent element so that ^e O, e

R

need not have an identity). As in

[2],

the two-sided Peirce decomposition of

R

relative ^to the idempotent e takes the form

R eReeR(l-e)(l-e)Re(l-e)R(l-e). ^We

will forma- e

Re

m,n 2 we may write

R RII +

lly set

el=

^{e and}

e2=

^{l-e So letting}

Rmn

^{m n}

Rmn

^{will be denoted by x}

RI2R21R22. ^Moreover

^an element of the subring

mn

From

the definition of d we note that

d(O) d(O0) d(O)O + Od(O)

O. Moreover, we have d(e) d(e

2)

d(e)e

+ ed(e). So

we can express d(e) as

all ⁺ a12 ⁺ a21 ⁺ a22

and use the value of d(e) to get that

all ^a22,

^{that is,}

all

⁰

a22.

Consequently, we have d(e)

a12

⁺

a21.

Now

let f be the inner derivation of R determined by the element

a12 a21,that

is

f(x) [x,al2 a21]

^{for all x in}

^R.

^Therefore,

^f(e) [e,al2 ^a21] a12

⁺

^a21.

616 M.N. DAIF

In

the sequel, and without loss of generality, we can replace the multiplicative derivation d by the multiplicative derivation d f, which we denote by D,that is,

D

d f. This yields

D(e) O.

This simplification is of great importance, for, as we will see, the subrings

R

become invariant under the multiplicative derivation

mn

D.

2. A

KEY LEMMA.

LEIA

I.

D(Rmn)Rmn,

^{m,n 1,2}

PROOF. Let

Xll

^be an arbitrary element of

RII.

^Then

^D(Xll) D(exl]e)=eD(Xll)e

which is an element of

RII. ^For

^{an element}

x12

ⁱⁿ

RI2,

^{we have}

D(Xl2) D(eXl2) eD(Xl2) bl]

⁺

b12. ^But

⁰

^{D(O) D(Xl2e) D(Xl2)e bll,}

^hence

^D(Xl2) b12

which belongs to

RI2. ^In

^{a similar} fashion, for an element

x21

ⁱⁿ

R21,

^{we have}

D(x21)

belongs to

R21. ^Now

^{take an element}

x22

ⁱⁿ

R22.

^Write

^D(x22 Cli+C12+c21+c22 ^So,

0

D(ex22) eD(x22) _Cll

⁺

c12,

^whence

Cll c12 ^O.

^Likewise

c21

^{O, and thus}

D(x22) c22

^{which is an element of}

R22.

^{This proves the 1emma.}

3.

CONDITIONS

OF

MARTINDALE.

In

his note

[l],

Martindale has given the following conditions which are imposed on a ring

R

having a family of idempotent elements

{ei:

^iI

(I) xR

0 implies x O.

(2)

If

e.Rx

0 for each i in

I,

then x 0 (and hence

Rx

0 implies x

0).

(3) For

each i in

I eixeiR(l-ei

^{0 implies e}i^xei ⁰

In

our note, we find it appropriate to simply dispense with conditions (i),

(2)

and

(3)

altogether and instead substitute the following conditions

(M l) ^xR

^{0 implies x O.}

(M 2) ^eRx

^{0 implies x 0 (and hence}

^Rx

0 implies x

0).

(M 3)

^{exeR(l-e) 0 implies exe}

^O.

4.

AUXILIARY LEIAS.

LEbIA 2.

For

any x in

R

and any x in

R

with p q, we have

mm mm pq pq

D(x

mm

+

xpq

D(Xmm) ⁺ D(Xpq).

PROOF. Assume

m p and q 2.

be an element of

R.I

^{n Using}

^Lemm

^we

Consider the sum

D(Xll)

⁺

D(Xl2) ^Let tln

D(x )t D(x

t

n x

D( D[(x +

x

)t

have

[D(Xll)

⁺

D(Xl2)]tln II In

11

tln II

12 n

llD(tln D(Xll

⁺

Xl2)tln ⁺ (Xll

⁺

Xl2)D(tln XllD(tln D(Xll ⁺ Xl2)tln"

^Thus,

[D(Xll)

⁺

D(Xl2) D(Xll

⁺

Xl2)]tln

^O.

In

the same fashion, for any

t2n

ⁱⁿ

R2n,

^{we can get the following}

[D(Xll) ⁺ D(Xl2) D(Xll

⁺

Xl2)]t2n

^O.

Combining these results, we have

[D(Xll)

⁺

D(x12) D(Xll

⁺

Xl2)]R

^O.

^By

^condition

(MI),

^{we obtain}

D(Xll ⁺ x12) D(Xll)

⁺

D(Xl2).

In

view of the symmetry resulting from ^condition

(bl I)

^{and the} implication of condition

(M2),

^{we can} find that the other three cases are easily shown in a similar fashion.

LEMMA

3. D is additive on

RI2.

PROOF. Let x12

^and

YI2

^{be two elements in the subring}

^R

12’ ^{and consider the sum}

WHEN IS A MULTIPLICATIVE DERIVATION ADDITIVE? 617

D(x12)

⁺

D(Y12).

D(x +

y

)t

(A)

For

an element

tln

ⁱⁿ

R1n,

^{we have}

[D(x12)

⁺

D(Y12)]tln

12 12

In’

since each side is zero by

Lemma

I, so

[D(x12) ⁺ D(Y12) D(x12

⁺

Y12)]tln ^O.

(B) Consider an element

t2n

ⁱⁿ

R2n. ^We

^have

^(x12

⁺

Y12)t2n

^(e

⁺ x12)( t2n

⁺

Y12t2n ^).

^Thus,

D[(Xl2 ⁺ Y12)t2n]

^D(e

⁺ x12)(t2n

⁺

Y12t2n ^{+ (e}

⁺

x12)D(t2n ⁺ _Y12t2n

=(D(e)

+

D(Xl2))(t2n

⁺

_Yl2t2n

^{+ (e +}

Xl2)(D(t2n)

⁺

D(Yl2t2n)) D(Xl2)t2n ⁺ Xl2D(t2n

+

D(Yl2t2n ^)’

^by

^Lemmas

^{and 2. Thus,}

D((Xl2

⁺

Yl2)t2n) D(Xl2t2n) ⁺ D(Yl2t2n). ^But (D(Xl2)

⁺

m(Yl))t

2n D(x 2

t2n +

m(y12

)t2n

^D(x

2t2n

^{+ m(y}

12t2n ^)-(x 12+Y12 ^)D( t2n ⁾⁼

D((Xl2

⁺

Yl2)t2n) (x12

⁺

Yl2)D(t2n) D(Xl2 ⁺ Yl2)t2n. ^Hence, [D(Xl2)

⁺

D(YI2) D(Xl2 ⁺ Yl2)]t2n

Consequently, from

(A)

and

(B)

we have

=0.

[D(Xl2) ⁺ D(Yl2) D(xl2

⁺

Yl2)]R ^O.

By

condition (M

1),

^{we have}

D(Xl2 ⁺ _Y12 D(Xl2) ⁺ D(Yl2).

LEMbIA 4. D is additive on

Rll.

PROOF. Let Xll

^and

Yll

be arbitrary elements in

Rll. ^For

^{an element}

t12

ⁱⁿ

R12,

we have

(D(X]l)

⁺

D(Yll))tl2-- D(Xll)tl2

⁺

D(Yll)tl2 D(Xlltl2) ⁺ D(Ylltl2) (Xll

⁺

Y11)D(t12). ^But _x11t12

^and

_Y11t12

^{are in}

RI2,

^and

^D

^{is additive on}

R12

^by

^Lemma

^3,

hence

(D(Xll)

⁺

D(Yll))tl2 D(Xlltl2

⁺

_Ylltl2 (Xll ⁺ Yll)D(tl2) D((Xll+Yll)tl2 (Xll ⁺ Yll)D(tl2) D(Xll

⁺

Yll)tl2.

^{thus we have}

[D(Xll) ⁺ D(Yll) D(Xll ⁺ Yll)]tl2

^=0.

Therefore,

[D(x11)

⁺

D(Y11) D(x11 ⁺ _Yll )]R12 ^O.

From Lemma I, D(Xll) ⁺ D(Yll D(Xll ⁺ _Yll

^{is an element in}

RII,

^{hence the above}

result with condition (M

3)

^give

D(Xll

⁺

_Yll D(Xll

⁺

D(Yll ^)"

LEMbIA 5.

D

is additive on

R11 ⁺ R12 ^eR.

PROOF.

Consider the arbitrary elements

xll, Yll

ⁱⁿ

RII

^and

^{x12, Y12}

ⁱⁿ

^{R12. So,}

Lemmas

_2,3,4 give

D((Xll

⁺

x12)

⁺

(Yll

⁺

_YI2

⁾⁾

D((Xll ⁺ _Yll

⁺

(x12

⁺

YI2))=D(Xll

⁺

Yll

⁺

D(Xl2

⁺

Y12

^D(x11

+ D(Y11)

⁺

D(Xl2)

⁺

D(Y12) (D(Xll) ⁺ D(x12)) ⁺ (D(Y11)

+

D(YI2)) D(Xll

⁺

x12)

⁺

D(Yll ⁺ _Y12

^{). Thus}

^D

^{is additive on}

RII ⁺ RI2.

the desired result.

This proves

5. MAIN

THEOREM.

THEOREM. Let R

be a ring containing an idempotent e which satisfies conditions

(MI),

^(M

2)

^and

(M3).

^{If d is} any multiplicative derivation of

R,

then d is additive.

PROOF.

As we mentioned before, and without loss of generality, we can replace d by

D. Let

x and y be any elements of

R.

Consider D(x) + D(y). Take an element t in eR

RII

⁺

RI2.

^{Thus, tx and ty are} elements of

eR.

According to

Lemma

5, we can obtain

t(D(x)

+ D(y))

tD(x)

+ tD(y) D(tx)

+

D(ty)

D(t)(x +y)

D(tx + ty)-

D(t(x

+ y))

618 N.N. DAIF

+ tD(x +

y).

Thus,

t(D(x)

+ D(y)) tD(x

+ y).

Since t is arbitrary in

eR,

we obtain

eR(D(x) +

D(y) D(x + y))

O.

By condition

(M2),

^{we get}

D(x +

y) D(x)

+

D(y),

which shows that the multiplicative derivation D is additive.

ACKNOWLEDGEMENT.

The author is indebted to the referee for his helpful suggestions and valuable comments which helped in appearing the paper in its present shape

REFERENCES

MARTINDALE

II1 k khen are Multiplicative Mappings Additive

Proc

Ame Math

Soc.

21 (1909), 695-69.

2.

JACOBSON, N. Structure

of Rings,

Amer.

blath.

Soc. Colloq.

Publ.

3_7 ^(1964).

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