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

This creation to noncommutative noetherian jewelry, obtainable to someone with a simple 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 comprises the elemental varieties of quantum teams.

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In particular, if K is a ﬁeld, this shows that A1 (K) is the K-algebra presented by two generators x, y and one relation xy − yx = 1. Definition. Let K[y1 , . . , yn ] be a polynomial ring in n independent indeterminates over a ring K. The formal partial derivatives ∂/∂y1 , . . , ∂/∂yn are commuting derivations on K[y1 , . . , yn ], and so we may form the iterated diﬀerential operator ring K[y1 , . . , yn ][x1 , . . , xn ; ∂/∂y1 , . . , ∂/∂yn ]. This ring is called the nth Weyl algebra over K and is denoted An (K).

Gn yt , and so k[G] = j=1 g∈Gn kgyj = t j=1 k[Gn ]yj , that is, k[G] is ﬁnitely generated as a left k[Gn ]-module. 5, and by symmetry it is right noetherian as well. • SIMPLICITY IN SKEW-LAURENT RINGS • We conclude the chapter with a few considerations about (two-sided) ideals in skew polynomial and skew-Laurent rings. In the theory of commutative noetherian rings, ideals are ubiquitous, but this is no longer true in the noncommutative theory, because noncommutative noetherian rings need not have very many ideals.

Let K[y1 , . . , yn ] be a polynomial ring in n independent indeterminates over a ring K. The formal partial derivatives ∂/∂y1 , . . , ∂/∂yn are commuting derivations on K[y1 , . . , yn ], and so we may form the iterated diﬀerential operator ring K[y1 , . . , yn ][x1 , . . , xn ; ∂/∂y1 , . . , ∂/∂yn ]. This ring is called the nth Weyl algebra over K and is denoted An (K). In view of Exercise 2G, if K is a right (left) noetherian ring, then all the Weyl algebras An (K) are right (left) noetherian.