By K. R. Goodearl, R. B. Warfield Jr

This advent to noncommutative noetherian jewelry, available to someone with a easy history in summary algebra, can be utilized as a second-year graduate textual content, or as a self-contained reference. broad explanatory fabric is given, and routines are built-in all through. New fabric comprises the elemental kinds of quantum teams.

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14, there is an index n such that Im ∩ S = In ∩ S for all m ≥ n. Thus Im = (Im ∩ S)T = (In ∩ S)T = In for all m ≥ n, which establishes the ACC for right ideals of T . Therefore T is right noetherian. The left noetherian case is proved symmetrically. From this corollary we immediately obtain that all quantum tori Oq ((k × )n ) are noetherian. Exercise 1R. 14. Let S = R[x; α] and T = R[x±1 ; α], where α is an automorphism of R. Show that the rule β(s) = x−1 sx deﬁnes an automorphism β of S, and then use the universal mapping property of S[y; β] to show that the inclusion map S → T extends to a ring homomorphism φ : S[y; β] → T such that φ(y) = x−1 .

Since Gi−1 is a normal subgroup of Gi , we have xGi−1 x−1 = Gi−1 , and hence x(k[Gi−1 ])x−1 = k[Gi−1 ]. As a result, the rule α(r) = xrx−1 deﬁnes an automorphism α of k[Gi−1 ]. By deﬁnition of α, we have xr = α(r)x for all r ∈ k[Gi−1 ], and thus k[Gi ] = k[Gi−1 ][x±1 ; α]. 15 now shows that k[Gi ] is noetherian. Thus, by induction, we conclude that k[Gn ] is noetherian. Now G is a t ﬁnite union of cosets Gn y1 , . . , Gn yt , and so k[G] = j=1 g∈Gn kgyj = t j=1 k[Gn ]yj , that is, k[G] is ﬁnitely generated as a left k[Gn ]-module.

I. Finally, let S = Sn = R[x1 ; δ1 ][x2 ; δˆ2 ] · · · [xn ; δˆn ]. The standard notation is S = R[x1 , . . , xn ; δ1 , . . , δn ]. (b) State and prove a universal mapping property for S analogous to Exercise 2F(a). (c) Show that S = R[xπ(1) , . . , xπ(n) ; δπ(1) , . . , δπ(n) ] for any permutation π of the index set {1, . . , n}. 30 CHAPTER 2 • WEYL ALGEBRAS • A fundamental class of skew polynomial rings is formed by taking differential operator rings over polynomial rings with respect to the standard (ordinary or partial) derivatives, as follows.