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The concept of nil-symmetric rings has been introduced as a generalization of symmetric rings and a particular case of nil-semicommutative rings. A ring

Throughout this paper, all rings are associative with unity. Given a ring

For a ring

A ring

let

Then any nilpotent element is a linear combination of

Let

Similarly by considering the opposite ring of

Clearly every symmetric ring is nil-symmetric but the converse is not true by Example

For a reduced ring

Let

From the above example we observe that a nil-symmetric ring need not be Abelian, as

An Abelian ring also need not be either a right nil-symmetric or a left nil-symmetric ring as shown by the following example.

We consider the ring in [

Let

Let

Let

Let

For a ring

For every reduced ring

Let

We also observe that every right (left) nil-symmetric ring is nil-semicommutative.

Every right (left) nil-symmetric ring is nil-semicommutative.

Let

The converse is however not true, as shown by the following example.

For every reduced ring

We have

Semicommutativity and nil-symmetry do not follow each other. In Example

Let

For a reduced ring

Let

Since the class of nil-symmetric rings is contained in the class of nil-semicommutative rings, the results which are valid for nil-semicommutative rings are also valid for nil-symmetric rings. Mohammadi et al. [

Let

By [

Finite product of right (left) nil-symmetric rings is right (left) nil-symmetric.

It comes from the fact that

Let

It suffices to prove the necessary condition because subrings of right (left) nil-symmetric rings are also right (left) nil-symmetric. Let

For a ring

It directly follows from Proposition

Let

It suffices to prove the necessary condition because subrings of right (left) nil-symmetric rings are also right (left) nil-symmetric. Let

Since the class of right (left) nil-symmetric rings is closed under subrings, therefore, for any right (left) nil-symmetric ring

Let

But for

For any nonempty subsets

A ring

Let

The following result shows that, for a semiprime ring, the properties of reduced, symmetric, reversible, semicommutative, nil-semicommutative, and nil-symmetric rings coincide. Note that a ring

For a semiprime ring

(1)–(4) are equivalent by [

Given a ring

For a reduced ring

Let

Considering the above proposition one may conjecture that if a ring

Let

Let

Homomorphic image of a right (left) nil-symmetric ring need not be a right (left) nil-symmetric ring. This is discussed after Example

Anderson-Camillo [

Let

The above example also helps in showing that homomorphic image of a right (left) nil-symmetric ring need not be a right (left) nil-symmetric ring. This is verified as follows.

In Example

Now we study some conditions under which the answer may be given positively. Since every right (left) nil-symmetric ring is nil-semicommutative by Proposition

We use the ring in [

If

Let

If

Let

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors are deeply indebted to Dr. Pierre-Guy Plamondon, Laboratory of Mathematics, University of Paris, France, for providing Example