^{1, 2}

^{1}

^{1}

^{2}

We introduce right (left) g-semisymmetric ring as a new concept to generalize the well-known concept: symmetric ring. Examples are given to show that these classes of rings are distinct. They coincide under some conditions. It is shown that

Throughout this paper, all rings are associated with identity and all modules are unitary. For a subset

(3) A right

(4) A right

(5) A left

(6) A left

Every symmetric ring is right g-semisymmetric ring, the converse is not true as illustrated by the following example, due originally to Bell [

(1) A ring

The following conditions are equivalent for a right

All cyclic submodules of

The following conditions are equivalent for a ring

Every right

Every cyclic right

If a ring

A one-sided ideal

Let

(1) The class of right g-semisymmetric rings is closed under subrings.

(2) The class of bounded right g-semisymmetric rings with boundaries 1 from left and 1 from right is closed under direct products.

(3) A ring is semiperfect and bounded right g-semisymmetric with boundary 1 from left if and only if

(4) A ring

(1) Trivial.

(2) Assume that

(3) Assume that

Conversely, suppose that

(4) By Lemma

Let

If

If

(1) Suppose

(2) Let

Suppose that

Let

Since

Now we are ready to prove the following proposition.

Flat left modules over bounded left g-semisymmetric ring with boundaries 1 from left and 1 from right are bounded left g-semisymmetric with boundaries 1 from left and 1 from right.

Let

In the following propositions

A torsionless

The following conditions are equivalent.

Every torsionless left

Every submodule of a free left

There exists a faithful, bounded g-semisymmetric left

An application of Propositions

For an

The left

If the left

If the right

An application of Proposition

Let

reduced if

ZI (zero-insertive ring) if

Let

if

if

If

Assume that

The previous lemma is false for rings without identity. Indeed, the ring

A ring

Let

Let

Since every reduced ring is symmetric, bounded right g-semisymmetric ring with boundary 1 from left and IFP, since every bounded right g-semisymmetric ring with boundary 1 from left is abelian, by Lemma

Let

Let

Suppose that

(1) if

(2) if

(3) if

hence

(5) if

(6) if

These cases prove that R is bounded right g-semisymmetric ring with boundary 2 from left and right.

The following example gives a bounded right g-semisymmetric ring with boundary 2 from left and right which is not symmetric.

Since

This Project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. 03-41/429. The authors, therefore, acknowledge with thanks DSR technical and financial support. The authors would like to express their great appreciation to the professor Mohammed Hessain Fahmy for helpful discussions, They had with him, during the preparation of this work. The authors thank the referee for his or her comments, which considerably improved the presentation.