By K. R. Goodearl, R. B. Warfield Jr
This advent to noncommutative noetherian jewelry, obtainable to a person with a easy heritage 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 contains the elemental forms of quantum teams.
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Additional resources for An Introduction to Noncommutative Noetherian Rings
A) Show that formal matrix addition and multiplication make sense in , and that by using those operations S0 B becomes a ring. T S B 0 T (b) Show that there is also a ring and that T 0 B S ∼ = S B 0 T T 0 B S of formal lower triangular matrices, . (c) Observe that the set 0B 0 0 of matrices 0 b 00 is an ideal of and that, under the obvious abelian group isomorphism of B onto 0B 0 0 S B 0 T , , left S-submodules (right T -submodules, (S, T )-sub-bimodules) of B correspond precisely to left ideals (right ideals, two-sided ideals) of S0 B contained in T 0B 0 0 .
Let K[y] be a polynomial ring over an arbitrary ring K, and let d/dy be the standard derivation on K[y]. The formal diﬀerential operator ring K[y][x; d/dy] is called the (ﬁrst) Weyl algebra over K and is denoted A1 (K). ) The Weyl algebra A1 (K) is generated (as a ring) by the elements of K together with x and y, which commute with the elements of K and satisfy the equation xy = yx + 1. Rather than adapt the terminology of presentations to include “rings presented over subrings,” let us phrase this description of A1 (K) in terms of a universal property.
The corresponding multiparameter quantum torus is ±1 the k-algebra Oq ((k × )n ) presented by generators x±1 1 , . . , xn and relations × n xi xj = qij xj xi for all i, j. The single parameter version Oq ((k ) ), for q ∈ k × , is the special case when qij = q for all i < j. Exercise 1P. Show that any quantum torus is an iterated skew-Laurent ring in parallel with Exercise 1K. A FEW NOETHERIAN RINGS 17 • A SKEW HILBERT BASIS THEOREM • We derive a version of the Hilbert Basis Theorem for the skew polynomial rings R[x; α] discussed above; an analogous result for skew-Laurent rings will follow as a corollary.
An Introduction to Noncommutative Noetherian Rings by K. R. Goodearl, R. B. Warfield Jr