Abstract
In this chapter we shall make a study of rings satisfying certain ascending chain conditions. In the non-commutative case-and this is really the only case with which we shall be concerned- the decisive and incisive results are three theorems due to Goldie. The main part of the chapter will be taken up with a presentation of these.
Definition. An element a in the ring R is regular if it is neither a left nor right zero divisor in R.
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Bibliography
L. Lesieur and R. Croisot : “Sur la decomposition en ideaux primaires dans un anneau non necessairement commutatif” Comptes Rendus Ac. Sc. Paris 243 (1965), pp. 1988–1991.
————“ Theory Noetherienne des anneaux, des demi-grou-pes et des modules dan le cas non commutatif I, II, III in Collo-que d'algebre superieure, Bruxelles 1956, Math. Annalen 134 (1958), pp. 458–476 and Acad. Roy. Belg. Bull. CI. Sci. (5) 44 (1958), pp. 75–93 respectively.
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Herstein, I.N. (2010). Topics In Ring Theory. In: Herstein, I.N. (eds) Some Aspects of Ring Theory. C.I.M.E. Summer Schools, vol 37. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11036-8_2
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DOI: https://doi.org/10.1007/978-3-642-11036-8_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-11035-1
Online ISBN: 978-3-642-11036-8
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