By K. R. Goodearl, R. B. Warfield Jr
This advent to noncommutative noetherian earrings, obtainable 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. vast explanatory fabric is given, and routines are built-in all through. New fabric contains the elemental forms of quantum teams.
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Extra resources for An Introduction to Noncommutative Noetherian Rings
B) If R is simple and no positive power of α is inner, show that the only nonzero ideals of S are S, Sx, Sx2 , . . (c) Now assume only that R is α-simple and no positive power of α is inner. Show that every nonzero ideal of S contains a power of x. ] 1ZA. Let R be a ring and α an automorphism of R. (a) Construct a skew power series ring R[[x; α]] whose elements are formal ∞ i power series i=0 ri x with coeﬃcients ri ∈ R, where xr = α(r)x for all r ∈ R. (b) Construct a skew Laurent series ring R((x; α)) consisting of formal ∞ Laurent series i=n ri xi with n ∈ Z and coeﬃcients ri ∈ R, where xr = α(r)x for all r ∈ R.
With this change, analogs of Steps 1–5 are easily carried out; we leave the details to the reader. A more eﬃcient way to deal with the switch from left-hand to right-hand coeﬃcients is to work with opposite rings – see Exercise 1Q. 14 are that the quantum planes Oq (k 2 ) are noetherian and (by induction) the quantum n-spaces Oq (k n ) are noetherian. A FEW NOETHERIAN RINGS 19 Exercise 1Q. Let α be an automorphism of a ring R. Show that α−1 is an automorphism of the opposite ring Rop and that R[x; α]op = Rop [x; α−1 ].
If a0 = 0, then px−1 = xn−1 + an−1 xn−2 + · · · + a1 would be a nonzero element of I ∩ S with degree n − 1, contradicting the minimality of n. Hence, a0 = 0. Observe that xpx−1 = xn + α(an−1 )xn−1 + · · · + α(a0 ), and so xpx−1 − p is an element of I ∩ S with degree at most n − 1. The minimality of n implies that xpx−1 − p = 0, and thus α(ai ) = ai for all i. Next, consider an arbitrary element r ∈ R, and note that pr = αn (r)xn + an−1 αn−1 (r)xn−1 + · · · + a0 r αn (r)p = αn (r)xn + αn (r)an−1 xn−1 + · · · + αn (r)a0 .
An Introduction to Noncommutative Noetherian Rings by K. R. Goodearl, R. B. Warfield Jr