Advances in Pure Mathematics, 2012, 2, 39-40

http://dx.doi.org/10.4236/apm.2012.21009 Published Online January 2012 (http://www.SciRP.org/journal/apm)

A Note on the (Faith-Menal) Counter Example

R. H. Sallam

Mathematics Department, Faculty of Science, Helwan University, Cairo, Egypt

Email: rsallams@hotmail.com

Received September 1, 2011; revised October 15, 2011; accepted October 22, 2011

ABSTRACT

Faith-Menal counter example is an example (unique) of a right John’s ring which is not right Artinian. In this paper we

show that the ring T which considered as an example of a right Johns ring in the (Faith-Menal) counter example is also

Artinian. The conclusion is that the unique counter example that says a right John’s ring can not be right Artinian is

false and the right Noetherian ring with the annihilator property rl(A) = A may be Artinian.

Keywords: John’s Ring; Artinian and Noetherian Rings

1. Introduction

A ring R is called right John’s ring if it is right Noethe-

rian and every right Ideal A of R is a right annihilator i.e.

rl(A) = A for all right ideals A of R.

John’s ([1], Theorem 1) by using a result of Kurshan

([2], Theorem 3.3), showed that a right Noetherian ring is

right Artinian provided that every right ideal is a right

annihilator.

Ginn [3] showed that Kurshan result was false. Ginn’s

example does not provide a counter example to John’s

theorem. Therefore the validity of John’s theorem was

doubtful.

Faith-Menal counter example proved that there is an

example (unique) of a right John’s ring which is not right

Artinian.

Here in this paper we prove the false of the Faith-

Menal counter example by proving that the considered

non-Artinian right john’s ring is in fact right Artinian. So

the John’s theorem may be true see [1].

All rings considered in this paper are associative rings

with identity.

We recall the Faith-Menal counter example in Section

1 and we prove that it is false in Section 2.

2. Section 1: The Counter Example

Example 8.16 (Faith-Menal) [4]. Let D be any countable,

extentially closed division ring over a field F, and let R =

DF

F

TRD

(x). Then T(R, D) is a non-Artinian right John’s ring.

Proof

Cohn shows that R is simple, principle right ideal do-

main that is right V ring (Theorems 8.4.5 & 5.5.5 [4])

and D is an R-R bimodule such that DR is the unique

simple right R module. Hence T(R, D) is a right John’s

ring by Theorem 8.15 (in this book). But T(R, D) is not

Artinian because if it were then R would also right

Artinian and hence a field which is a contradiction.

Here

and T0,d R

For more information about this example see [4].

3. Section 2: A Note on the Counter Example

Theorem

The right John’s ring T(R, D) defined in the counter ex-

ample is Artinian.

Proof

Recall the following (Exercises 10.7 [5]):

Let φ: D→R be a ring homomorphism and let M be a

right R-module (or left R-module) then

1) Via φ M is right D-module;

2) If MD is Artinian or Noetherian then so is MR;

3) If R is finite dimensional algebra (via φ) over a field

D then the following is equivalent:

a) MR is Artinian and Noetherian

b) MR is finitely generated

c) MD is finite dimension

1) Consider the ring homomorphism φ: D → R defined

by φ (d) = d

1f. Every R-module homomorphism is a

D-homomorphism via φ. Since D is a division ring so it

will be semisimple ring and hence every right D-module

is semisimple (Corollary 8.2.2 [3]).

2) If the ring TRD

is John’s ring (as in the coun-

ter example above) then it is Noetherian and hence R and D

are Noetherian. As D is right Noetherian ring then every

finitely generated right D module is Noetherian (6.1.3 [6]).

3) Every finitely generated right D-module M is semi-

simple so it is right Artinian.

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