By K. R. Goodearl, R. B. Warfield Jr
This advent to noncommutative noetherian jewelry, available to an individual with a uncomplicated history 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 comprises the elemental varieties of quantum teams.
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Additional info for An Introduction to Noncommutative Noetherian Rings
Other cases were subsumed in work on skew group rings. 17 as we have stated it was proved by Jordan in his dissertation  but not published until [1984, Theorem 1]. 2. Skew Polynomial Rings We continue the study of skew polynomial rings in this chapter, ﬁrst discussing the case where the multiplication is twisted by a derivation, and then developing the general case. Since our main motivation for looking at skew polynomial rings is to be able to construct and work with further important examples of noetherian rings, most of the later part of the book could be read independently of this chapter.
Xn ] as a polynomial ring in the single indeterminate xn with coeﬃcients from the ring R[x1 , . . , xn−1 ]. 10. Let R be an algebra over a ﬁeld k. If R is commutative and ﬁnitely generated as a k-algebra, then R is noetherian. Proof. Let x1 , . . , xn generate R as a k-algebra, and let S = k[y1 , . . , yn ] be a polynomial ring over k in n independent indeterminates. Since R is commutative, there exists a k-algebra map φ : S → R such that φ(yi ) = xi for each i, and φ is surjective because the xi generate 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. We have already noted that S always has some nontrivial ideals, namely Sx, Sx2 , .
An Introduction to Noncommutative Noetherian Rings by K. R. Goodearl, R. B. Warfield Jr