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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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