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

16 1. A DICTIONARY ON RINGS AND IDEALS
Proof. Let (a) be irreducible and x, y /∈ (a). Then (a, x), (a, y) are principal ideals
properly containing (a) giving (a, x) = (a, y) = R. Let ba + cx = da + ey = 1
and look at (ba+cx)(da+ey) = 1 to see that xy /∈ (a). It follows that (a) is prime,
part (1) of 1.5.3. If (b) is not irreducible, then (b) = (a1)(b1) for some irreducible
(a1) and (b) ⊂ (b1). Continue this process to get (b) = (a1) . . . (an)(bn) for some
irreducibles (a1), . . . (an). The chain of ideals (b) ⊂ (b1) ⊂ · · · ⊂ (bn) has a
union
n (bn) which is a principal ideal. The generator of the union must be in
some (bn). Therefore (bn) = (bn+1) giving that (bn) is irreducible. This gives a
factorization required in 1.5.3 part (2).
1.5.7. Example. The integers Z is a principal ideal domain and therefore a unique
factorization domain.
1.5.8. Definition. The supremum of a set of elements in P is the greatest common
divisor and an infimum is the least common multiple.
1.5.9. Corollary. In a unique factorization domain the greatest common divisor
and the least common multiple of a finite set of elements exist.
If (a) = (p m1
1 ) . . . (pmk k ) and (b) = (pn1 1 ) . . . (pnk k ) with mi, ni ≥ 0 then greatest
common divisor is (p min(m1,n1)
1 ) . . . (p min(mk,nk)
k ) and least common multiple is
).
(p max(m1,n1)
1
) . . . (p max(mk,nk)
k
1.5.10. Exercise. (1) Show that an irreducible element in a principal ideal domain generates
a maximal ideal.
(2) Show that there are infinitely many prime numbers.
(3) Let Z[ √ −1] be the smallest subring of C containing √ −1. Show that Z[ √ −1] is a
principal ideal domain.
(4) Let Z[ √ −5] be the smallest subring of C containing √ −5. Show that Z[ √ −5] is not
a unique factorization domain.
1.6. Polynomials
1.6.1. Definition. Let R be a ring. The polynomial ring R[X] is the additive
group given by the direct sum
n RXn , n = 0, 1, 2, . . . consisting of all finite
sums f = a0 + a1X + . . . amX m , a polynomial with an ∈ R being the n ′ th
coefficient. Multiplication is given by XiX j = Xi+j extended by linearity. If
g = b0 + b1X + . . . bnX n is an other polynomial, then
f + g = (a0 + b0) + (a1 + b1)X + · · · + (ak + bk)X k + . . .
fg = a0b0 + (a0b1 + a1b0)X + · · · + (a0bk + a1bk−1 + · · · + akb0)X k + . . .
A monomial is polynomial of form aX n . The construction may be repeated to
give the polynomial ring in n-variables R[X1, . . . , Xn] or even in infinitely many
variables.
1.6.2. Definition. The degree, deg(f), of a polynomial 0 = f ∈ R[X] is the
index of the highest nonzero coefficient, the leading coefficient. A polynomial
with leading coefficient the identity is a monic polynomial.
1.6.3. Remark. (1) R is identified with the subring of constants in the polynomial
ring R[X1, . . . , Xn].
(2) The nonzero constants are the polynomials of degree 0.
(3) The constant polynomial 1 is the unique monic polynomial of degree 0 and
the identity in the polynomial ring.