Let

All rings considered in this note are nonzero commutative rings with identity. Unless otherwise specified, we consider rings

Let

Before we describe the results that are proved in this note, it is useful to recall the following definitions from [

For any

The set of vertices of

It is known that for any commutative ring

Let

It is useful to recall the following definitions from commutative ring theory before we proceed further. Let

Let

Let

Moreover, it was shown in [

For any set

Let

Let

Let

In Section

Suppose that

Suppose that

Let

Let

Let

Let

Let

We end this note with an example of an infinite ring

Let

Let

Let

The following lemma establishes some necessary conditions on

Let

Since

The next lemma is [

Let

For the sake of completeness, we give below an argument for the fact that

We provide in the next lemma some sufficient conditions on

Let

The proof of this lemma is contained in the proof of [

We know from Lemma

We next state and prove the main theorem of this section.

Let

It is well known that

If

Assume that

The following remark determines the center of

Let

If

Suppose that

We next present some examples to illustrate the results proved in this section.

Let

(i) Let _{1}^{c}_{1}^{c}

(ii) Let

Let

Let

For the sake of convenience we split the results proved in this section into several lemmas. We begin with the following lemma. We make use of this lemma in the proof of Lemma

Let

Suppose that for some

In the following lemma, we determine the girth of

Let

Let

Though the following lemma is elementary, we include it for the sake of future reference.

Let

As

The next lemma discusses the girth of

Let

If

If

By hypothesis,

(i) Assume that

Conversely, suppose that either

If

(ii) Suppose that

Without loss of generality, we may assume that

Thus in both the cases,

We know from [

We show in the next lemma that if

Let

Since by the assumption that

With the help of the above lemmas, we obtain the main theorem of this section.

Let

The proof of this theorem follows immediately from [

We next proceed to consider rings

Let

Note that

Suppose that

We next show that

Observe that if

Thus if

If

The following remark characterizes rings

Let

Since

Let

We determine in the following remark rings

Let

We are assuming that

We next have the following corollary, the proof of which is immediate from the results proved in this section.

(i) Let

(ii) Let

Let

We first prove some elementary lemmas which are of interest in their own right and which are useful in proving the main results of this section. We begin with the following lemma.

Let

If

If

(i) Suppose that

(ii) Let

Using Lemma

Let

If

If

(i) Suppose that

(ii) Let

We next study in the following corollary to Lemma

Let

If

If

(i)(a) Let

(i)(b) If

(ii)(a) Let

Suppose that

(ii)(b) This can be proved using similar arguments as in the proof of (ii)(a) and using Lemma

The following proposition is one among the main results in this section. We show in this proposition that if a ring

Let

(i)

(ii)

(iii)

(i)

(ii)

(iii)

Let

Let

We first recall the following facts from [

Let

This fact is easy to check. The relation

For an element

Recall from [

The following fact is important, and we make use of it in the proof of Lemma

Let

The Fact

It is well known that any element of a von Neumann regular ring can be expressed as the product of a unit and an idempotent [

Note that if

Now it follows from the above two preceding paragraphs that given any

Let

Let

Let

Let

Indeed, it is true that if

This proves that the subgraph of

We include the following simple lemma for the sake of completeness.

Let

The only idempotent elements in a field are 1 and 0. Using this observation, it follows that

Let

Let

Let

We know from the proof of Lemma

Let

Let

Let

Let

Let

From (

We make use of the following useful remark in Example

Let

Let

We next have the following example.

(i) Let

Let

Let

(i) As any nilpotent element of

(ii) Suppose that

(iii) We now verify that

By (i), we have

We have

This contradicts the hypothesis that

(iv) We obtain from (ii) that

Let

Let R be a Noetherian ring which is not an integral domain. Suppose that

(i)

(ii)

(iii)

Recall that a commutative ring

Let

Let

(i)

(ii)

The following remark determines

Let

As is already observed in the proof of (i)

(i) Suppose that

(ii) Suppose that ^{c}

We next mention an example to illustrate that in Remark

Let

As

Let

This shows that if

We next claim that the subgraph of

Let

Let

By hypothesis,

The next remark provides examples of rings

We remark here that Lemma

Let

Let

Let

We conclude this note with the following example of an infinite ring

Let

Let

We are assuming that

This proves that if

The author is very much thankful to the academic editors Professor David F. Anderson, Professor Vesselin Drensky, and Professor Dolors Herbera for their useful and valuable suggestions.