On Weak Symmetric Rings

Abstract

For a ring endomorphism α and an α-derivation δ, we introduce weak symmetric rings and weak (α, δ)-symmetric rings which are a generalization of symmetric rings, and investigate their properties. It is proved that: (1) If R is a (α, δ)-compatible and reversible ring, then R is weak symmetric if and only if R[x; α, δ] is weak symmetric; (2) If R is a semicommutative ring, then R is weak (α, δ)-symmetric if and only if R[x] is weak symmetric, where and are the extended maps of α, δ, respectively.

Key Words:

• (α, δ)-Compatible ring
• α-Rigid ring
• Weak (α, δ)-symmetric ring

2000 Mathematics Subject Classification:

• Primary 16S36
• Secondary 16S99

ACKNOWLEDGMENTS

This research is supported by Scientific Research Fund of Hunan Provincial Education Department (07c268), the National natural science foundation of China (10771058), Hunan Provincial Natural Science Foundation of China (06jj20053), and Scientific Research Fund of Hunan Province Education Department (06A017). The authors thank the referee for his (her) careful reading and valuable comments which improve the presentation of this article.

Notes

Communicated by T. Albu.