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

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

The already wide diversity of purposes of ring concept has been greater 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 line with Groebner bases to check them. This publication develops those equipment in a self-contained manner, targeting an in-depth learn of the idea of an enormous category of non-commutative jewelry (encompassing so much quantum groups), the so-called Poincaré-Birkhoff-Witt earrings. We contain algorithms which deal with crucial elements like beliefs and (bi)modules, the calculation of homological size and of the Gelfand-Kirillov size, the Hilbert-Samuel polynomial, primality exams for top beliefs, etc.

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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. 23. Let M be a left R-module. The following assertions are equivalent: (a) M is left artinian and left noetherian; (b) M has finite length.

Therefore, necessarily r n = O for some n < N(b). 30 we obtain r-gcd(a,b) = r-gcd(ro,rd ~ r-gcd(rl,r2). 38 1. GENERAUTIES ON RINGS If r2 = O then n = 2 and we are done. 30 yields r-gcd(a, b) = r-gcd(b, r2) ~ ... ~ r-gcd(rn-I. r n ) = rn-I. o which is the desired result. 32 (Bezout's Theorem). Let R be a euclidean domain and let a and b be non-zero elements of R. If 9 = r-gcd(a, b), then there exist elements u, v E R such that 9 = ua+vb. PROOF. Let UI 1 O Ui Ui-2 - qi-l Ui-l Vo VI O 1 Vi Vi-2 - qi-l Vi-l Uo for 2:$;i:$;n for 2:$;i:$;n + Vi-2 b ) - qi-l (Ui-I a and By induction on i, it is easy to see that ri = = ri-2 - qi-l ri-l = (Ui-2 a (Ui-2 - qi-Iui-l)a + (Vi-2 - qi-IVi-db = Ui a + Vi-I b ) + Vi b andhence r-gcd(a, b) = rn-l = Un-la + Vn-Ib.

On the other hand, for each positive integer n with O ::; n ::; p - 1, the chain stabilizes, say at qn. If we choose = max{p,Ql,Q2, ... ,qp-d, then we obtain Kin = Kmn for every positive integer n and every m i;:::m. , the above ascending chain of left S-ideals stabilizes and Sis left noetherian, indeed. The right noetherian case may be proved similarly. 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.

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