| Summary: | 博士 === 國立中央大學 === 數學系 === 105 === In 1966, P. M. Cohn introduced the concept of a generalized Euclidean ring, inspired by the property that any invertible matrix over a Euclidean ring can be row-reduced to the dentity matrix by elementary matrices. In 1984, Dennis, Magurn and Vaserstein proved that the integral group ring ZCm of finite cyclic group Cm is generalized Euclidean.
It is well known that a Euclidean ring is quasi-Euclidean and a quasi-Euclidean ring is generalized Euclidean. In this thesis, we construct a finitely generated nonprincipal ideal of ZG for nontrivial abelian group G to show that ZG is neither Euclidean nor quasi-Euclidean. Moreover, we give an upper bound for the number of generators of an ideal in ZCm. The case m being a power of a prime is treated more seriously. In the final chapter, following the Wedderburn-Artin Theorem, we give a more accessible proof than Bass' to show that a semilocal ring has stable rank one, hence it is a generalized Euclidean ring.
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