Commutative algebra - Department of Mathematical Sciences - old ...

20 1. A DICTIONARY ON RINGS AND IDEALS
1.9. Power series
1.9.1. Definition. Let R be a ring. The power series ring R[[X]] is the additive
group
n RXn , n = 0, 1, 2, . . . of all power series anX n with n ′ th coefficient
an ∈ R. Multiplication is given by X i X j = X i+j extended by linearity. For
another power series bnX n the sum and product are
anX n + bnX n = (an + bn)X n
anX n · bnX n = (
k
an−kbk)X n
The construction may be repeated to give the power series ring in n-variables
R[[X1, . . . , Xn]] or even in infinitely many variables.
1.9.2. Remark. The polynomial ring is identified as a subring R[X] ⊂ R[[X]] of
power series with only finitely many nonzero terms.
1.9.3. Definition. The order, o(f), of a power series 0 = f ∈ R[[X]] is the index
of the least nonzero coefficient.
1.9.4. Proposition. If R is a domain, then R[[X]] is a domain and for 0 = f, g ∈
R[X]
o(fg) = o(f) + o(g)
Proof. Clearly the lowest nonzero coefficient in the product is the product of the
two lowest nonzero coefficients.
1.9.5. Proposition. A power series f = anX n is a unit if and only if a0 is a
unit.
Proof. It suffices to look at a power series f = 1 − gX. Then the power series
1/f = 1 + gX + g 2 X 2 + · · · + g n X n + . . . is well defined and f · 1/f = 1.
1.9.6. Proposition. If K is a field, then K[[X]] is a principal ideal domain. and
(X) is the only nonzero prime ideal.
Proof. If the lowest order of an element in an ideal I is n. Then clearly I =
(X n ).
1.9.7. Corollary. If K is a field, then K[[X]] is a unique factorization domain.
1.9.8. Proposition. Let I ⊂ R be an ideal. Then there is a canonical surjective
homomorphism
R[[X]]/IR[[X]] → R/I[[X]]
1.9.9. Corollary. If Q ⊂ R[[X]] is a maximal ideal, then P = Q ∩ R ⊂ R is a
maximal ideal and Q = (P, X).
Proof. X ∈ Q so R/Q ∩ R R[[X]]/Q.
1.9.10. Exercise. (1) Show that the ring Z[[X]] is not a principal ideal domain.
(2) Show that the ring Q[[X, Y ]] is not a principal ideal domain.
(3) Let K be a field. Show that (X1, . . . , Xn) is the unique maximal ideal in the power
series ring K[[X1, . . . , Xn]].
(4) Let a ∈ R be nilpotent. Show that the ring R[[X]]/(X − a) is isomorphic to R.
(5) What is R[[X]]/(X − a) if a ∈ R is a unit?
(6) Let I ⊂ R be an ideal. Show that IR[[X]] ⊂ R[[X]] is not a maximal ideal.