By Nicolas Bourbaki
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Additional resources for Algebra II: Chapters 4-7 (Pt.2)
It is clear that bn is an ideal in the ring A [[XI], generated by the monomials of degree n. Hence b, consists of finite sums of products of n formal power series without constant terms ; if D is a derivation of A[[X]], we have whence it follows at once that Db, c b,-l for n 3 1. Since the sequence (b,)n,o is a fundamental system of neighbourhoods of 0 in A[[X]] (IV, p. 26 remark), D is continuous and (i) is proved. 34 POLYNOMIALS AND RATIONAL FRACTIONS 04 Let A be a derivation of A [XI into A [[XI].
Q ( x I + x 2 ) @factors) To prove (iii), let Gpl,,,,,pn be the set of permutations of (1, p, restrictions to the intervals + . - - + p,) whose are increasing. By I, p. 60, example 2 and Prop. 2, (ii) we have (with pi factors xi) and this sum is equal to In (iii) let us put n = 2, x, = x, = x, p, = q and p, = r, then we obtain (iv) (I, loc. ). Finally (v) follows from Prop. 2, (ii) and Prop. 2 of I, p. 100, applied to the elements xi of the ring T(M). v Remarks. N(') put - 1) Let (xi)i If (Xi) E A(') and p p ,, be a family of elements of M.
The family (TS"(M)), , is a graduation of type N of the algebra TS(M), and the unit element of T ( M ) is a unit element of TS(M). Let G,, be the set of those (T E G p t q such that , The mapping (u, T ) H U T of G , , x 6,,, into 6,+, is bijective (I, p. 60, q Example 2) ; hence if z E TSP (M) and z' E TS (M), we have PROPOSITION 2. - (i) The A-algebra T S ( M ) is associative, commutative and unital. , pn be integers > 0, and let GP,I ,,, u E 6 ~ , + . . , z, Z1 Z2 . Z n E TS'" ( M ) , then = %PI +..