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Standard prime ideals and lying over for finite extensions of Noetherian algebras. (English) Zbl 0840.16020
Let $$R$$ be a Noetherian algebra of finite Gelfand-Kirillov dimension $$\text{GK}(R)$$ over the field $$k$$. A standard prime factor series of the finitely generated right $$R$$-module $$M$$ is a finite sequence of submodules $$0=N_0\subset N_1\subset\cdots\subset N_n=M$$ such that for each $$i$$ $$P_i=r_R(N_i/N_{i-1})$$ is the unique associated prime of $$N_i/N_{i-1}$$ and $$\text{GK}(R/P_i)\leq \text{GK}(R/P_j)$$ whenever $$i\leq j$$. The set of primes of $$R$$ arising from such a series for $$M$$ is an invariant of $$M$$, called the set of standard primes, $$St(M)$$, of $$M$$. This concept was introduced and studied by G. Krause [in Abelian groups and noncommutative rings, Contemp. Math. 130, 215-229 (1992; Zbl 0772.16009)], where it was shown that $$St (M)$$ is the set of primes minimal over an annihilator of a nonzero submodule of $$M$$. In this paper $$St(M)$$ is studied when $$M$$ is an $$S$$-$$R$$-bimodule, finitely generated on each side, where $$S$$ is also a Noetherian $$k$$-algebra of finite $$\text{GK}$$-dimension. The maximal lengths of standard series of $$_SM$$ and $$M_R$$ are shown to be finite and equal, for example. These ideas are then applied to the situation where $$R \subseteq S$$ and $$S_R$$ is finitely generated. In particular, the author explores the relationship between standard primes and issues of Lying Over and Lying Directly Over – this concept having been introduced by K. R. Goodearl and E. S. Letzter [Prime ideals in skew and $$q$$-skew polynomial rings, Mem. Am. Math. Soc. 521 (1994; Zbl 0814.16026)].
##### MSC:
 16P90 Growth rate, Gelfand-Kirillov dimension 16P40 Noetherian rings and modules (associative rings and algebras) 16D20 Bimodules in associative algebras 16D25 Ideals in associative algebras
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##### References:
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