Abstract
This paper is an alternative and complement to [1]. First we take up the idea of an axiomatic notion of dimension generalizing and unifying Gelfand-Kirillov- and Gabriel-Rentschler-dimension. We introduce an “axiom of invariance”, generalizing an idea of Stafford. Next we apply this to reprove the main results of [1] on “good behaviour” of prime ideals in certain extension-rings of noncommutative rings, including an additivity principle for Goldie-ranks. Finally we discuss the extent to which our “restriction” on extensions is also necessary in order to have results of this type.
Preprint, Wuppertal, June 1981
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Bibliography
Borho, W.: On the Joseph-Small additivity principle for Goldie-ranks, IHES preprint No. 4, Jan. 1981.
Borho, W.—Gabriel, P.—Rentschler, R.: Primideale in Einhüllenden auflösbarer Lie-Algebren, Springer Lecture Notes in Math. 357 (1973).
Borho, W.—Kraft, H.: Über die Gelfand-Kirillov-Dimension, Math. Ann. 220 (1976), 1–24.
Brown, K.A.—Lenagan, T.H.—Stafford, J.T.: Weak ideal invariance and localization, J. London Math. Soc. 21 (1980), 53–61.
Brown, K.A.—Lenagan, T.H.—Stafford, J.T.: K-Theory and stable structure of Noetherian group rings, Proc. London Math. Soc., to appear.
Goldie, A.—Krause, G.: Artinian quotient rings of ideal invariant Notherian rings, J. of Algebra 63 (1980), 374–388.
Gordon, R.—Robson, J.C.: Krull dimension, Mem. AMS 133 (1973).
Joseph, A.—Small, L.W.: An additivity principle for Goldie rank, Israel J. Math. 31 (1978) 105–114.
Krause, G.—Lenagan, T.H.—Stafford, J.T.: Ideal invariance and Artinian quotient rings, J. Algebra 55 (1978), 145–154.
Small, L.W.: Orders in Artinian rings, J. Algebra 4 (1966), 13–41.
Stafford, J.T.: Stable structure of non-commutative Noetherian rings, J. Algebra 47 (1977), 244–267.
Stafford, J.T.: On the regular elements of Noetherian rings, Proc. 1978 Atwerp Conference, Marcel Dekker Co., New York.
Chatters, A.W.—Goldie, A.W.—Hajarnavis, C.R.—Lenagan, T.H.: Reduced Rank in Noetherian Rings, J. Algebra 61 11969), 582–589).
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© 1982 Springer-Verlag
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Borho, W. (1982). Invariant dimension and restricted extension of Noetherian rings. In: Malliavin, MP. (eds) Séminaire d'Algèbre Paul Dubreil et Marie-Paule Malliavin. Lecture Notes in Mathematics, vol 924. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0092925
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DOI: https://doi.org/10.1007/BFb0092925
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