• 검색 결과가 없습니다.

Addenda to

N/A
N/A
Protected

Academic year: 2022

Share "Addenda to"

Copied!
1
0
0

로드 중.... (전체 텍스트 보기)

전체 글

(1)

ADDENDA TO

"ON THE COMPACTNESS OF THE STRUCTURE SPACE OF A RING"

Wuhan Lee

1.

In

[1]

we gave necessary and sufficient condition for the compactness of the structure space of a ring with the property that

(C')

Every principal two-sided ideal is modular.

If

the zero ideal

(0)

is modular, then the ring A has an identity and com'crsely if A has an identity then every ideal is mcdu'ar. This factleads to that the condition (C') is simply equivalent to the condition that A has an identity and it is well known that for a ring with an identity the structure space is always compact.

As pointed out by

R.

Blair in his recent letter to us, our original intention was to state as follows ill€tead of (C'):

(C*) Every nonzero principal two-sided ideal is modular.

Proposition 7 and Theorem 2 in

[1]

with (C*) instead of

(C'),

however are still valid under the hypothesis that A is not a radical ring; since

B~

to}, B contains a nonzero principal two-sided ideal and hence B is modular.

2. Let consider a riug A with the property that

(C") No nonzero homomorphic image of A is a radical ring.

According to R. Blair and L.C. Eggan

[2],

for such a ring the structure space is compact

if

and only

if

A is generated as an ideal by a finite number of elements.

We shall show relations among those classses of rings satisfying conditions (C), (C'), (C"),

and (C*).

(C)

For every Ua of {Ua), aEA,Dua is moC:ular.

Proposition

1.

If A is not a radical ring then (C*) implies (C").

- 5

Proof: Let B be a proper ideal in A.

If

B=={o}, then

A/B:::::.A

is not a radical ring; and if

B~

{O}, then B is modular by(C*), so

A/B has an

identity and is therefore again not a radical ring.

The above result shows that Proposititon 7 in

(l)

follows immediately from the Corollary to Theorem 2 in (2J.

Proposition

2.

Condition (C) implies neither (C") nor (C*).

Proof: Let

Bbe

a simple ring witn an identity element nd let

R

be a nonzero radical ring, and let A be the direct sum BEE1R. Then A has exactly one primitive ideal, namely, R. Thus, for any aEA, either Du.==A(if aER) or DUa==

R (if aER). In' either case, DUa is modular, so that A satisfies

(C).

However, A/B:::::.R is a radical ring, so

A

does not satisfy (C"). Mor- e:JVer A can have no identity element,

so

B

is

not modular. Thus A does not satisfy (C"').

Proposition

3.

Neither

(C")

nor

(C*)

implies

(C).

Proof: Let A

be

a simple ring without an ide- ntity element. Clearly A satisfies

(C")

and (C*).

The only primitive ideal is the zero ideal {O}.

If

aEA with

a~O,

the fuJ=={O}. Since A has no identity element, DUa is not modular,

so

A does not satisfy

(C).

REFERENCES

1.

Wuhan Lee,

On The Compactness of The Stluclure Space of A Ring,

J. Korean Math.

Soc.

vol. 1 (1964)

p.

3-5

2. R.

Blair and L.C. Eggan.

On The Compact- ness of The StfucluI'c Space of A Ring,

Proe. Amer. Math. Soc. vol. 11 (1950)

pp. 876-879.

참조

관련 문서

The “Asset Allocation” portfolio assumes the following weights: 25% in the S&P 500, 10% in the Russell 2000, 15% in the MSCI EAFE, 5% in the MSCI EME, 25% in the

Advisor: prof. We will look into a criterion of definition and type of condition the base of a existing theory. Expression called condition is defined

(Attached to a verb) This is used to show that the succeeding action takes place right after the preceding action.. (Attached to a verb) This is used to indicate a

With the opening of the Songdo Bridge 4, IFEZ is looking forward to less congestion in traffic that connects from Songdo International City locations, including Incheon New

 When a symmetry operation has a locus, that is a point, or a line, or a plane that is left unchanged by the operation, this locus is referred to as the symmetry

* When a symmetry operation has a locus, that is a point, or a line, or a plane that is left unchanged by the operation, this locus is referred to as the symmetry element..

Utilizing the notion that the variational operator behaves like a differential operator, we proceed to establish the necessary condition for an extremum... Thus the boundary

Because it is more difficult to design PID controllers for a MIMO process than for a SISO process that there is an interaction between the different