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

ISBN-10: 9048163285

ISBN-13: 9789048163281

ISBN-10: 9401702853

ISBN-13: 9789401702850

The already large diversity of functions of ring idea has been improved within the eighties by way of the expanding curiosity in algebraic constructions of substantial complexity, the so-called type of quantum teams. one of many primary homes of quantum teams is they are modelled by way of associative coordinate earrings owning a canonical foundation, which permits for using algorithmic constructions according to Groebner bases to check them. This ebook develops those tools in a self-contained method, targeting an in-depth learn of the concept of an unlimited category of non-commutative jewelry (encompassing so much quantum groups), the so-called Poincaré-Birkhoff-Witt jewelry. We contain algorithms which deal with crucial features 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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**Sample text**

8(1) s; 1. 13. LetJ be a two-sided ideal of R[x; a, 8]. If ] () R is astable, then it is also 8 -stable. LEMMA If rE] () R, then both xr and a (r) x belong to]. It thus obviously follows that 8(r) = xr - a(r)x E ] () R, which proves the assertion. D PROOF. 14. 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) ~ ...

11. 11] shows that the assumptions in the previous result cannot be omitted. Let 1<. [x]. In this case S = R [y; a] is clearly not a domain, as EXAMPLE yx = xa(y) = o. On the other hand, S is not prime either, since ySx s; yxS = O. Note also that since IiSXyi and IixiyS are easily seen to actually be direct sums, S is not noetherian on either side. Let us condude this section with some fust comments about ideals of Ore extensions. So, let us fix some quasi-derivation (a, 8) on R. 12. A (le(t, right or two-sided) ideal I of R is said to be a-stable resp.

It appears that the associated ring ([[x; 0"] is an instance of a more general construction, which we will describe and study in the present section. 1. Let O" be a ring endomorphism of R. A O"-derivation of R is an additive endomorphism 8 : R -+ R with the property that 8(rs) = 0"(r)8(s) + 8(r)s for all r, 5 E R. We will call the pair (O", 8) a quasi-derivation. DEFINITION For example, if O" is an arbitrary ring endomorphism of R, then we may associate to anya ERa O"-derivation 8, by putting 8(r) = ar O"(r)a, for alI r E R.

### Algorithmic Methods in Non-Commutative Algebra: Applications to Quantum Groups by José Bueso, José Gómez-Torrecillas, Alain Verschoren (auth.)

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