properties of nil and nilpotent ideals

properties of nil and nilpotent ideals

Lemma 1

.

Let

A

⊂

B

be ideals of a ring

R

. If

A

is nil and

B

/

A

is nil, then

B

is nil. If

A

is

nilpotent

and

B

/

A

is nilpotent, then

B

is nilpotent.

Proof.

Suppose that

A

and

B

/

A

are nil. Let

x

∈

B

. Then

x

n

∈

A

for some

n

, since

B

/

A

is nil. But

A

is nil, so there is an

m

such that

x

n

⁢

m

=

(

x

n

)

m

=

0

. Thus

B

is nil.

Suppose that

A

and

B

/

A

are nilpotent. Then there are

natural numbers

n

and

m

such that

A

m

=

0

and

B

n

⊆

A

. Therefore,

B

n

⁢

m

=

0

.
∎

Lemma 2

.

The sum of an arbitrary family of nil ideals is nil.

Proof.

Let

R

be a ring, and let

ℱ

be a family of nil ideals of

R

. Let

S

=

∑

I

∈

ℱ

I

. We must show that there is an

n

with

x

n

=

0

for every

x

∈

S

. Now, any such

x

is actually in a sum of only finitely many of the ideals in

ℱ

. So it suffices to prove the lemma in the case that

ℱ

is finite. By

induction

, it is enough to show that the sum of two nil ideals is nil.

Let

A

and

B

be nil ideals of a ring

R

. Then

A

⊂

A

+

B

, and

A

+

B

/

A

≅

B

/

(

A

∩

B

)

, which is nil. So by the first lemma,

A

+

B

is nil.
∎

Lemma 3

.

The sum of a finite family of nilpotent left or

right ideals

is nilpotent.

Proof.

We prove this for right ideals. Again, by induction, it suffices to prove it for the case of two right ideals.

Let

A

and

B

be nilpotent right ideals of a ring

R

. Then there are natural numbers

n

and

m

such that

A

n

=

0

and

b

m

=

0

.

Let

k

=

n

+

m

-

1

. Let

z

1

,

z

2

,

…

,

z

k

be elements of

A

+

B

. We may write

z

i

=

a

i

+

b

i

for each

i

, with

a

i

∈

A

and

b

i

∈

B

. If we expand the

product

z

1

⁢

z

2

⁢

⋯

⁢

z

k

we get a sum of terms of the form

x

1

⁢

x

2

⁢

…

⁢

x

k

where each

x

i

∈

{

a

i

,

b

i

}

.

Consider one of these terms

x

1

⁢

x

2

⁢

⋯

⁢

x

k

. Then by our choice of

k

, it must contain at least

n

of the

a

i

’s or at least

m

of the

b

i

’s. Without loss of generality, assume the former.
So there are indices

i

1

<

i

2

<

…

<

i

n

with

x

i

j

∈

A

for each

j

.
For

1

≤

j

≤

n

-

1

, define

y

j

=

x

i

j

⁢

x

i

j

+

1

⁢

⋯

⁢

x

i

j

+

1

-

1

,
and define

y

n

=

x

i

n

⁢

x

i

n

+

1

⁢

⋯

⁢

x

k

.
Since

A

is a right ideal,

y

j

∈

A

.

Then

x

1

⁢

x

2

⁢

⋯

⁢

x

k

=

x

1

⁢

x

2

⁢

⋯

⁢

x

i

1

-

1

⁢

y

1

⁢

y

2

⁢

⋯

⁢

y

n

∈

x

1

⁢

x

2

⁢

⋯

⁢

x

i

1

-

1

⁢

A

n

=

0

.

This is true for all choices of the

x

i

, and so

z

1

⁢

z

2

⁢

⋯

⁢

z

k

=

0

. But this says that

(

A

+

B

)

k

=

0

.
∎

Title

properties of nil and nilpotent ideals

Canonical name

PropertiesOfNilAndNilpotentIdeals

Date of creation

2013-03-22 14:12:54

Last modified on

2013-03-22 14:12:54

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

6

Author

mclase (549)

Entry type

Result

Classification

msc 16N40

Related topic

KoetheConjecture

Related topic

NilIsARadicalProperty

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