By José Bueso, José Gómez-Torrecillas, Alain Verschoren (auth.)

The already wide variety of functions of ring conception has been superior within the eighties through the expanding curiosity in algebraic buildings of substantial complexity, the so-called type of quantum teams. one of many basic homes of quantum teams is they are modelled by means of associative coordinate earrings owning a canonical foundation, which permits for using algorithmic constructions in response to Groebner bases to check them. This publication develops those tools in a self-contained manner, focusing on an in-depth research of the suggestion of an unlimited type of non-commutative earrings (encompassing such a lot quantum groups), the so-called Poincaré-Birkhoff-Witt earrings. We contain algorithms which deal with crucial facets like beliefs and (bi)modules, the calculation of homological size and of the Gelfand-Kirillov measurement, the Hilbert-Samuel polynomial, primality exams for high beliefs, etc.

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**Extra resources for Algorithmic Methods in Non-Commutative Algebra: Applications to Quantum Groups, 1st Edition**

**Sample text**

Let (a, 8) be a quasi-derivation on R and let I be a astable two-sided ideal of R. If a is an automorphism and if R satisfies the ascending chain condition on two-sided ideals, then a (1) = 1. LEMMA 3. ORE EXTENSIONS PROOF. 27 Dur assumptions imply the ascending chain 1~ (]"-l (1) ~ (]"-2 (1) ~ ... , there exists some positive integer n with the property that (]"l-n(1) = (]"-n(1). Then = (]"n«(]"l-n(1)) = (]"n«(]"-n(1)) = 1, (]"(1) o since (]" is bijective. 15. Let «(]", 8) be a quasi-derivation on R such that (]" is an automorphism and let 1 be a two-sided ideal of R which is both (]" -stable and 8 -stable.

It is fairly easy to see that any left artinian and left noetherian left R-module has finite length. Indeed, a composition series for M may be constructed inductively as follows. Put No = O, and let NI be a left R-submodule of M strictly containing No and minimal as such - this exists, since M is left artinian. We then choose N2 to be a minimal left R-submodule of M, which strictly contains NI, and so ono This chain O = No C NI C N2 C ... ), since M is left noetherian and, by its very definition, it yields a composition series for M.

Let R be a PID. Then for any r E R, which is neither zero nor invertibIe, the Ie{t R -module R / Rr has finite Iength. An interesting class of PIDs is given by the so-called euclidean domains. 27. :::; N(rs) for every 0'* r, 5 E R; (2) for every pair of elements a and b '* O in R, there exist q, r E R with the property that a = qb + r, where r = O or N(r) < N(b). Left euclidean domains are defined similarIy. If R is euclidean both on the right and the Ie{t, then we just say that R is a euclidean domain.