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Sep 16, 1986 - other hand, in a nil ring, every element is trivially a product of nilpotents. ... multiplicatively generated by its nilpotents is nil if it is Artinian or if it.
Internat. J. Math. & Math. Sci. (1988) 5-8 VOL. ii NO.

A NOTE ON RINGS WHICH ARE MULTIPLICATIVELY GENERATED BY IDEMPOTENTS AND NILPOTENTS HAZAR ABU-KHUZAM Dp_rtment of Mathematical Sciences University of Petroleum and Minerals Dhahran, Saudi Arabia

(Received June 2, 1986 and in revised form September 16, 1986)

ABSTRACT.

We give the structure of certain rings which are multiplicatively

generated by nilpotents or multiplicatively generated by idempotents and nilpotents.

KEY WORDS AND PHRASES. Boolean ring, Nil ring, polynomial identy. 1980 AMS SUBJECT CLASSIFICATION CODE. 16A70. 1.

INTRODUCTION. On the

In a Boolean ring, every element is trivially a product of idempotents.

other hand, in a nil ring, every element is trivially a product of nilpotents.

This

motivates the study of the structure of a ring, which as a seml-group, is generated

by its idempotents, or is generated by its nilpotents, or more generally, is Indeed, we prove that a ring which is

generated by its idempotents and nilpotents.

multiplicatively generated by its nilpotents is nil if it is Artinian or if it m+l m f(x) (f(x) is a polynomial with integer x satisfies the polynomial identity x

coefficients).

We also prove that if

R

is a ring which is multiplicatively

generated by its idempotents and nilpotents such that the set N

elements is commutative, then

R

forms an ideal of

N

R/N

and

of nilpotent

We

is Boolean.

also give examples to show that our conditions are essential for the validity of our theorems.

We start with the following definitions, the first of which was introduced in [I].

DEFINITIONS.

A ring

by its idempotents.

R

A ring

generated by its nilpotents.

is called an 1-ring if as a semigroup

R

R

is called an N-ring if as a semi-group

R

is generated

R

is said to be an Nl-ring if as a semlgroup

is

R

is

generated by its idempotents and nilpotents.

I].

The following two theorems were proved in

THEOREM A.

Let

R

be an 1-ring with identity.

THEOREM B.

Let

R

be a finite 1-ring.

Then

Then

R

R

i__s Boolean.

i__s Boolean.

REMARKS. i.

A homomorphic image of an 1-ring, N-ring, or an Nl-ring is an 1-ring, N-ring, or

an Nl-ring.

R

R

{0}.

2.

If

3.

Trivially, every 1-ring and every N-ring is an Nl-ring.

is an N-ring with identity, then

H. ABU-KHUZAM

6

An 1-ring need not be Boolean as shown in [I].

4.

An N-ring need not be nil (see

An Nl-ring need not be neither Boolean nor nil (see Example 2

below).

Example

below).

MAIN RESULTS.

2.

In preparation for the proofs of our theorems, we start with the following lemmas. Lemma

is known but we give its proof for completeness.

LEMMA 1.

xm(f(x)) TM PROOF.

(f(x))

m

R

Let

f(x)

polynomial

x

x

m+l

If a

J.

x

m

H

for all

xm+2

xf(x)

xm(f(x)) m

e

x

Then

R.

i__n

f(x)).

m

x

of

R

is nil.

x

xm+If (x)

m

xm(f(x)) TM is an idempotent element in m 0 for (f(x) )m (Lemma I), we obtain x

then the

By Lemma I, x

m

x

2m

2m

is an idempotent.

rin J

x

Continuing we get

satisifes the polynomial identity

0 and since

So

m

R

R

x e J.

Let

PROOF.

in

f(x)

which implies that

Jacobson radical

x (f(x))

idempotent of

is an

m

LEMMA 2.

m

be a ring such that for some positive integer m, and some m m+l x f(x) for all x i__n R.

with integer coefficients, x

J.

So

every

x

is nil.

In the following two

In [lJ, it is proved that a finite 1-ring is Boolean.

Indeed, we prove that an N-ring R m m+l x satisfies the polynomial identity x f(x).

theorems we study the analogous case for N-rings. is nil of

R

PROOF.

R

is Artinian or if

THEOREM I.

R

Let

Let

J

be an Artinian

R/J

{0}, by Remark 2.

J, and hence

R

THEOREM 2.

(m i_s

R.

R

R

Let

be an

f(x)

R

nilpotent.

is

N-rin

is a

R/J

J # R, then

Suppose

(being

is an N-ring with identity (Remark I).

J # R.

This contradicts our assumption that

So

J

is nilpotent, since

positive and

R/J

So

semisimple Artinian) has an identity.

Thus

Then

N-rin.

be the Jacobson radical of

is nilpotent in an Artinian ring. m+l m x f(x) satisfying th___e polynomial identity x

polynomial with integer coefficients).

Then

R

is

nil.

PROOF.

By Lemma 2, the Jacobson radical J of R is nil. R/J being semisimple Each nonR/J is a subdirect product of prime rings R

is semiprime, and hence

R

zero prime ring

of [2], R

has a nontrivial center. m

By Lemma i, e idempotent of

m

satisfies the identity

ca(f(ca))

Ra. ea

#

Let

x

xm+if (x)

# 0

c

m

is an idempotent of m 2m 0, otherwise

and hence by Theorem 1.4.2

R

be a central element of

R

and hence

ca ca (f(ca))m

0

is a central

e

which contradicts the

fact that c a is a nonzero central element of a prime ring and cannot be a zero So divisor by Lemma 2.1.3 of [3]. But e R e x R x 0 for all x a a a a a a e x x 0 for all x and hence R has an identity element. So in R is an N-ring (Remark I) with identity.

R/J

{0}, and R

J

So

R

0

(Remark 2).

R

This implies that

is nil.

We now give an example to show that Theorem

need not be true if

R

is not

Artinian and Theorem 2 need not be true if R does not satisfy the identity m m+l x x f(x). The ring used in the following example was used in [I] to show that an 1-ring need not be Boolean.

EXAMPLE I. x

D

Let

matrices over

be any element of

D

R

be any ring with identity, and let

be the ring of all

in which at most a finite number of entries are nonzero.

R.

Then, for some positive integer

n

and some

nxn

Let

matrix

7

RINGS GENERATED BY IDEMPOTENTS AND NILPOTENTS A

D

over

we have

A

Let

are zero matrices.

n

n

S

O’s

nxn,

is

O’s

A

T

0

are zero matrices.

0

S

It is easy to verify that

is not nilpotent.

is an N-ring which is not nil since

ST.

X

are nilpotent elements, and

T

and

Thus

R

This example shows

0

0

or the hypothesis

that we cannot drop the hypothesis that R is Artinian in Theorem m+l m x f(x) in Theorem 2. that R satisfies the identity x

The following example shows that

Next we study the structure of certain Nl-rings. an Nl-ring need not be neither Boolean nor nil.

Let R

Example 2.

GF(2).

over

is a finite Nl-ring which is neither Boolean nor nil.

R

Trivially,

In example 2 above, the Nl-ring R

elements forms an ideal of

R

potent elements of

R

If

N

R

is an ideal of

R

So

N

R

2

exe)

exe) (xe ex

2

of nilpotent elements

is multiplicatively generated

has nonzero idempotents.

e(xe

Let

Clearly, (ex

R.

exe) (ex

exe)

e

be any nonzero

exe)

N

and

0.

0, and hence

-exexe

2 ex e.

Using induction, (I) implies that

2

n

(2)

(exe)

Let

a e N.

(3)

eae e N

Since

ea

N eae

ex

2

n e

for all positive integers

(2)

Then using

for every

a

N

and

ae

n.

we obtain

N.

N is a subring of eae e N we get

is commutative,

o__f

is Boolean.

is commutative, and the theorem

N

is nil since

So we may assume that

This implies that

(exe)

R

has no nonzero idempotents, then

e(ex

N

R/N

and

idempotent of R and let x be any element of exe) e N. Now, since N is commutative (xe

(i)

of nilpotent

This motivates the study in the

is Boolean.

be an Nl-ring such that the set

Then

by nilpotents only. follows.

R/N

N

commute.

Le__t R

THEOREM 3.

is commutative.

PROOF.

has the property that the set

Indeed, we prove that an Nl-ring will have this property if the nil-

next theorem.

R

and

R

R.

So using (3) and the fact that

8

H. ABU-KHUZAM ea e N, ae

(4)

R

Now since

N

N

Let

x

+ N

that either e

en.

2,

x

N

R.

be any nonzero element of

x e N

e

e.

(4) implies that

is an ideal of

x

and every idempotent

is multiplicatively generated by idempotents and nilpotnets and since

is commutative,

(5)

a e N

for every

x

or

e

e

en

2

R/N.

R

Since

is an Nl-ring, (5) implies

for some idempotent elements

So e

e

en + N

2

(el

+ N)

+ N)...(e n + N),

(e2

and hence

R/N

(6) If

is an I-ring.

R/N, then

is any idempotent element of

R/N.

nilpotent elements of for all

xe

x

ex

exe

(7)

The idempotents of

is Boolean.

R/N

R/N

Now, by (6) and (7),

But in

R/N

R/N

(

)

and

(

e)

has no nonzero nilpotent elements.

are

Thus

and hence

are central.

R/N

is 1-ring with central idempotent elements, and hence

This completes the proof of Theorem 3.

We now give an example to show that Theorem 3 need not be true if the nilpotents of

R

do not commute.

EXAMPLE 3. Nl-ring.

Let

Clearly,

R

te

be the ring of Example i. set

N

Then

of nilpotent elements of

R, being an N-ring, is an R

is not an ideal of

R.

This example shows that we cannot drop the hypothesis that the nilpotent commute in

Theorem 3.

REFERENCES

I.

PUTCHA, M.S. and YAQUB, A.

Idempotents,"

"Rings which are Multiplicatively Generated by Communications in Alsebra 8(1980), 153-159.

2.

HERSTEIN, I.N.

3.

HERSTEIN, I.N. Noncommutative Rinss, The Carus Math. Monographs, John Wiley and Sons, Inc., 1968.

Rings With Involution, University of Chicago Press, Chicago, 1976.