By K. R. Goodearl, R. B. Warfield Jr
This advent to noncommutative noetherian jewelry, available to someone with a easy historical past 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 workouts are built-in all through. New fabric comprises the elemental forms of quantum teams.
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Extra resources for An Introduction to Noncommutative Noetherian Rings
B) No positive power of α is an inner automorphism of R. Proof. As noted above, Exercises 1T and 1U show the necessity of conditions (a) and (b). Conversely, assume that (a) and (b) hold. 22 CHAPTER 1 Let I be a nonzero ideal of T ; we must show that I = T . 15 that I = (I ∩S)T . Thus, I ∩S = 0. Since I is an ideal in T , we are allowed to multiply it by either x or x−1 . Thus xIx−1 ⊆ I and x−1 Ix ⊆ I, whence xIx−1 = I. We also have xSx−1 = S (cf. Exercise 1R), and therefore x(I ∩ S)x−1 = I ∩ S.
Let T = R[x±1 ; α], where α is an automorphism of R. Suppose that some positive power αn is an inner automorphism of R; say there is a unit u ∈ R such that αn (r) = uru−1 for all r ∈ R. Set v = uα(u)α2 (u) · · · αn−1 (u), and show that α(v) = v. ] Next, show that 2 v −1 xn lies in the center of T . ] Finally, show that 1 + v −1 xn is not invertible in T , and conclude 2 that T (1 + v −1 xn ) is a proper, nonzero (two-sided) ideal of T . A skew polynomial ring S = R[x; α] has no chance to be simple, since Sx is always a nontrivial ideal of S (as are Sx2 , Sx3 , .
Yn and each αi is the automorphism of K sending yi to yi + 1 and ﬁxing the other yj . Quantum Planes. 2]. Skew-Laurent Rings. The skew Laurent series ring Q(t)((x; α)), where α is the automorphism of Q(t) sending t to 2t, was constructed by Hilbert to show the existence of a noncommutative ordered division ring [1903, Theorem 39]. Noetherian Skew Polynomial Rings. Finite generation of left ideals for skew polynomial rings in several variables over a ﬁeld was proved by Noether and Schmeidler [1920, Satz III].