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

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

This advent to noncommutative noetherian earrings, obtainable to a person with a easy historical past in summary algebra, can be utilized as a second-year graduate textual content, or as a self-contained reference. huge explanatory fabric is given, and routines are built-in all through. New fabric comprises the elemental forms of quantum teams.

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N ], where K consists of (C∞ ) functions in variables y1 , . . , 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].

Similarly, once Si has been constructed for some i < n, construct Si+1 = Si [xi+1 ; δˆi+1 ], where δˆi+1 is the unique derivation on Si such that δˆi+1 |R = δi+1 and δˆi+1 (xj ) = 0 for j = 1, . . , 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 ]. 1X. Let S = R[x; α] and T = R[x±1 ; α], where α is an automorphism of R. Suppose that α is inner; say there is a unit u ∈ R with α(r) = u−1 ru for all r ∈ R. Show that S = R[ux], an ordinary polynomial ring (that is, ux is a central indeterminate). Similarly, show that T = R[(ux)±1 ], an ordinary Laurent polynomial ring. 1Y. Show that R[x±1 ; α]op = Rop [x±1 ; α−1 ] if α is an automorphism of a ring R. (Cf. ) 1Z. 17 if the skew-Laurent ring T = R[x±1 ; α] is replaced by a skew polynomial ring S = R[x; α], where we continue to assume that α is an automorphism of R.

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