Ideal-Symmetric and Semiprime Rings

Abstract

Lambek extended the usual commutative ideal theory to ideals in noncommutative rings, calling an ideal A of a ring R symmetric if rst ∈ A implies rts ∈ A for r, s, t ∈ R. R is usually called symmetric if 0 is a symmetric ideal. This naturally gives rise to extending the study of symmetric ring property to the lattice of ideals. In the process, we introduce the concept of an ideal-symmetric ring. We first characterize the class of ideal-symmetric rings and show that this ideal-symmetric property is Morita invariant. We provide a method of constructing an ideal-symmetric ring (but not semiprime) from any given semiprime ring, noting that semiprime rings are ideal-symmetric. We investigate the structure of minimal ideal-symmetric rings completely, finding two kinds of basic forms of finite ideal-symmetric rings. It is also shown that the ideal-symmetric property can go up to right quotient rings in relation with regular elements. The polynomial ring R[x] over an ideal-symmetric ring R need not be ideal-symmetric, but it is shown that the factor ring R[x]/xⁿR[x] is ideal-symmetric over a semiprime ring R.

Key Words:

• Ideal-symmetric ring
• Matrix ring
• Polynomial ring
• Right quotient ring
• Semiprime ring
• Symmetric ring

2010 Mathematics Subject Classification:

• 16U80
• 16S70

ACKNOWLEDGMENTS

The authors thank the referee for his/her very careful reading of the manuscript and valuable suggestions that improved the paper. The second named author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (No. 2010-0022160). The third named author was supported by the Research Fund Program of Research Institute for Basic Sciences, Pusan National University, Korea, 2012, Project No. RIBS-PNU-2012-103.

Notes

Communicated by T. Albu.