The Structure of Finite c-Local Rings

Abstract

For a commutative ring R, assume that c is a nonzero element of Z(R) with the property that cZ(R) = {0}. A local ring R is called c-local if Z(R)² = {0, c}, Z(R)³ = {0}, and xZ(R) = {0} implies x ∈ {0, c}. For any finite c-local ring (R, 𝔪), it is proved that the ideal m has a minimal generating set which has a c-partition. The structure and classification up to isomorphism of all finite commutative c-local rings with order greater than 2⁵ are determined.

Key Words:

• c-Partition
• Finite c-local rings
• Minimal generating set
• Polynomial rings
• Structure

2000 Mathematics Subject Classification:

• 13M05
• 13M10

ACKNOWLEDGMENTS

This research is supported by the National Natural Science Foundation of China (Grant No. 10671122). The second author is also partly supported by a grant of Science and Technology Commission of Shanghai Municipality (STCSM No. 09XD1402500).

Notes

Communicated by I. Swanson.