On Generalized Euclidean Rings

• Main Authors: Abdul Rahman Tambunan, 湯何曼
• Other Authors: Ming-Guang Leu
• Format: Others
• Language: en_US
• Published: 2017
• Online Access: http://ndltd.ncl.edu.tw/handle/928hqd

碩士 === 國立中央大學 === 數學系 === 105 === In this thesis, a generalized Euclidean ring, or GE-ring for short, a notion introduced by P. M. Cohn are studied. Properties and examples of GE-rings and GE_n-rings but not GE-rings are derived. Following the result of Bass, stable rank of a ring R (denoted by sr(R)) is related to the general linear group over R. Every ring with stable rank one is a GE-ring. A principal ideal domain (ring) has stable rank ≤ 2. For a principal ideal domain R with stable rank one, R must be a Euclidean ring. Examples of GE-rings with stable rank higher than one are given. For the ring of integers O_K in the quadratic field K = Q(√d) with d a square free rational integer, sr(O_K) = 2.