On g-Semisymmetric Rings

Abstract EN

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 R is bounded right g-semisymmetric with boundary 1 from right if and only if R is symmetric, whenever R is regular.

It is shown that a ring R is strongly regular if and only if R is regular and bounded right g-semisymmetric with boundary 1 from right.

For a right p.p.-ring R it is shown that R is reduced if and only if R is symmetric, if and only if R is bounded right g-semisymmetric ring with boundary 1 from left, if and only if R is IFP, if and only if R is abelian.

We prove that there is a special subring of the ring of 3×3 matrices over a ring without zero divisors which is bounded right g-semisymmetric with boundary 2 from left and boundary 2 from right.

Also we show that 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.