An Introduction to Noncommutative Noetherian Rings by K. R. Goodearl, R. B. Warfield Jr

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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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].

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