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- W1995548195 abstract "In [I], Gabriel discussed a “Krull dimension” for rings and modules, but here we prefer to call it the Gabriel dimension, reserving the name Krull dimension for the notion defined in [7] and [4]. This paper is concerned with the class of those modules which have a Gabriel dimension. As some of the results presented here make clear, this class is large, strictly including the class consisting of all modules with Krull dimension and all modules over right noetherian rings. The initial stimulus for the investigation was our conjecture that, in a ring with Krull dimension, the prime radical is nilpotent. Now in Section 2 we obtain a detailed knowledge of the relationship between the Krull dimension and the Gabriel dimension of a module or, indeed, of an object in any Grothendieck category. This knowledge can be applied, as in [6] (see also [5] and [3]) to establish the above conjecture. However the methods used here lead, in [2, Section 51, to a more direct proof of this conjecture. They also help to answer other questions concerning modules with Krull dimension, see [2, Section 41. In the rest of this paper we study two other aspects of Gabriel dimension. In Section 3 we show that, when it exists, the Gabriel dimension of a commutative ring is virtually the same as its classical Krull dimension, as defined in [4]. Then in Section 4 we show that, unlike the case of Krull dimension, having a Gabriel dimension is inherited by a polynomial ring, and upper and lower bounds for this dimension are obtained. This resembles a result of Seidenberg [8] comparing the (finite) classical Krull dimension of a commu-" @default.
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- W1995548195 date "1974-06-01" @default.
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- W1995548195 title "The Gabriel dimension of a module" @default.
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- W1995548195 doi "https://doi.org/10.1016/0021-8693(74)90081-7" @default.
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