這篇論文研究由P. M. Cohn 所提出之 generalized Euclidean ring（簡稱 GE-ring）的概念。當中也介紹了GE-ring 與 GE_n-ring（非GE-ring）的例子與其性質。從 Bass 的一項結果：「一個環 R 的 stable rank（表示成 sr(R)）與 GLn(R) 有關」可以得知，所有 stable rank 為 1 的環會是一個 GE-ring。另外有一事實則是一個主理想整環（principal ideal domain） 的 stable rank 則會小於等於 2。若一主理想整環的 stable rank 為 1，則其必為一歐幾里得環。文中也給出一些 stable rank 大於 1 之 GE-ring 的例子。對於在二次體（quadratic field） K=(Q√d)（其中d為一無平方數因數 square free）中的所有整數環（ring of integers, O_K），可以得到 sr(O_K)=2。

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.

[1] M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, Reading, Massachusetts, 1969.

[2] H. Bass, K-theory and stable algebra, Inst. Hautes Etudes Sci. Publ. Math. 22 (1964), 5-60.

[3] W.-Y. Chang, On a paper of P. M. Cohn, Master thesis, National Central University, Chung-li, Taiwan, 2015.

[4] W.-Y. Chang, C.-R. Cheng, and M.-G. Leu, A remark on the ring of algebraic integers in Q(√d), Israel Journal of Mathematics 216 (2016), 605-616.

[5] H. Chu, On the GE_2 of graded rings, Journal of Algebra 90 (1984), 208-216.

[6] P. M. Cohn, On the structure of the GL_2 of a ring, Inst. Hautes Etudes Sci. Publ. Math. 30 (1966), 5-53.

[7] D. L. Costa, Zero dimensionality and the GE_2 of polynomial rings, Journal of Pure and Applied Algebra 50 (1988), 223-229.

[8] K. Dennis, B. Magurn, and L. Vaserstein, Generalized Euclidean group rings, J. Reine Angew. Math. 351 (1984), 113-128.

[9] D. S. Dummit and R. M. Foote, Abstract Algebra, 3rd ed., John Wiley and Sons, New York, 2003.

[10] S. C. Geller, On the GEn of a ring, Illionis J. Math. 21 (1977), 109-112.

[11] F. Grunewald, J. Mennicke, and L. Vaserstein, On the groups SL(Z[x]) and SL_2(k[x; y]), Israel Journal of Mathematics 86 (1994), 157-193.

[12] L. Guyot, On quotients of generalized Euclidean group rings, arXiv: 1604.08639[math.AC] (2016).

[13] A. J. Hahn and O. T. O'Meara, The Classical Groups and K-Theory, Springer-Verlag, New York, 1989.

[14] T. W. Hungerford, Algebra, Springer-Verlag, New York, 1974.

[15] T. Y. Lam, A First Course in Noncommutative Rings, Springer-Verlag, New York, 2001.

[16] T. Y. Lam, Exercises in Classical Ring Theory, Springer-Verlag, New York, 1995.

[17] M.-G. Leu, Lecture Notes, 2016.

[18] B. A. Magurn, An Algebraic Introduction to K-Theory, Cambridge University Press, Cambridge, 2002.

[19] H. Matsumura, Commutative Ring Theory, translated from the Japanese by M. Reid, 2nd ed., Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, Cambridge, 1989.

[20] C. P. Milies and S. K. Sehgal, An Introduction to Group Rings, Kluwer Academic Publishers, Dordrecht, 2002.

[21] P. Samuel, About Euclidean rings, Journal of Algebra 19 (1971), 282-301.

[22] A. A. Suslin, On the structure of the special linear group over polynomial rings, Mathematics of the USSR-Izvestiya, 11:2 (1977), 221-238.

[23] A. A. Suslin and L. N. Vaserstein, Serre's problem on projective modules over polynomial rings, and algebraic K-theory, Izv. Akad. Nauk SSSR Ser. Mat., 40 (1976), 937-1001.

[24] L. N. Vaserstein, Stable rank of rings and dimensionality of topological spaces, Funktsional. Anal. i Prilozhen. 5:2 (1971), 17-27.