Contents - Student subdomain for University of Bath

24 CHAPTER 1. INTRODUCTION
(i) ∀f, g ∈ I, f − g ∈ I,
(i) ∀f ∈ R, g ∈ I, fg ∈ I.
Notation 4 If S ⊂ R, we write (S) for ⋃ n∈N {∑ n
i=1 f ig i f i ∈ R, g i ∈ S}: the
set of all linear combinations (over R) of the elements of S. We tend to abuse
notation and write (a 1 , . . . , a k ) instead of ({a 1 , . . . , a k }). This is called the (left-)
ideal generated by S, and clearly is a left-ideal.
There are also concepts of right ideal and two-sided ideal, but all concepts agree
in the case of commutative rings. Non-trivial ideals (the trivial ideals are {0}
and R itself) exist in most rings: for example, the set of multiples of m is an
ideal in Z.
Proposition 1 If I and J are ideals, then I + J = {f + g : f ∈ I, g ∈ J} and
IJ = {fg : f ∈ I, g ∈ J} are themselves ideals.
Definition 8 A ring is said to be noetherian, or to satisfy the ascending chain
condition if every ascending chain I 1 ⊂ I 2 · · · of ideals is finite.
Theorem 1 (Noether) If R is a commutative noetherian ring, then so is R[x]
(Notation 10, page 30).
Corollary 1 If R is a commutative noetherian ring, then so is R[x 1 , . . . , x n ].
Definition 9 A ring R is said to be an integral domain if, in addition to the
conditions above, there is a neutral element 1 such that 1 ∗ a = a and, whenver
a ∗ b = 0, at least one of a and b is zero.
Another way of stating the last is to say that R has no zero-divisors (meaning
none other than zero itself).
Definition 10 An element u of a ring R is said to be a unit if there is an
element u −1 such that u ∗ u −1 = 1. u −1 is called the inverse of u.
Definition 11 If a = u∗b where u is a unit, we say that a and b are associates.
Proposition 2 If a and b are associates, and b and c are associates, then a
and c are associates. In other words, being associates is an equivalence relation.
For the integers, n and −n are associates, whereas for the rational numbers,
any two non-zero numbers are associates.
Definition 12 An ideal I of the form (f), i.e. such that every h ∈ I is gf for
some g ∈ R, is called principal. If R is an integral domain such that every ideal
I is principal, then R is called a principal ideal domain, or P.I.D.
Classic P.I.D.s are the integers Z, and polynomials in one variable over a field.
Inside a principal ideal domain, we have the classic concept of a greatest common
divisor.