本帖最后由 hbghlyj 于 2023-8-5 23:34 编辑
3. Prime and maximal ideals
3.1. Definitions and Examples.
Definition. An ideal $P$ in a ring $A$ is called prime if $P = A$ and if for every pair $x, y$ of elements in $A\setminus P$ we have $xy \not∈ P$. Equivalently, if for every pair of ideals $I,J$ such that $I,J \not⊂ P$ we have $IJ \not⊂ P$.
Definition. An ideal $\frak m$ in a ring $A$ is called maximal if $\mathfrak m = A$ and the only ideal strictly containing $\frak m$ is $A$.
Exercise.
(1) An ideal $P$ in $A$ is prime if and only if $A/P$ is an integral domain.
(2) An ideal $\frak m$ in $A$ is maximal if and only if $A/\frak m$ is a field.
Of course it follows from this that every maximal ideal is prime but not every
prime ideal is maximal.
Examples.
(1) The prime ideals of $\mathbb Z$ are $(0),(2),(3),(5),...$; these are all maximal except
$(0)$.
(2) If $A =\mathbb C[x]$, the polynomial ring in one variable over $\mathbb C$ then the prime
ideals are $(0)$ and $(x - λ)$ for each $λ ∈\mathbb C$; again these are all maximal
except $(0)$.
(3) If $A =\mathbb Z[x]$, the polynomial ring in one variable over $\mathbb Z$ and $p$ is a prime
number, then $(0), (p), (x),$ and $(p, x) = \{ap + bx|a, b ∈ A\}$ are all prime
ideals of $A$. Of these, only $(p, x)$ is maximal.
(4) If $A$ is a ring of $R$-valued functions on a set for any integral domain $R$ then
$I = \{f ∈ A|f(x)=0\}$ is prime.
Exercise. What are the prime ideals of $\Bbb R[X]$? What can you say about the prime
ideals of $k[X]$ for a general field $k$?
As we will see as the course goes on — and you might already guess from these
examples — prime ideals are central to all of commutative algebra.
In modern algebraic geometry the set of prime ideals of a ring $A$ is viewed as
the points of a space and $A$ as functions on this space. The following lemma tells
us that in this viewpoint a ring homomorphism $f : A → B$ defines a function from
the space associated to $B$ to the space associated to $A$. At first sight this reversal
of direction may seem perverse but it is one of those things we have to live with.
Suppose that $f : X → Y$ is a function then we may define a ring homomorphism
$f^∗ : R^Y → R^X$ by $f^∗(θ) = θ ∘ f$. Notice also, for example that if $f$ is continuous
then $f$ restricts to a ring homomorphism $C(Y ) → C(X)$.
The following lemma is attempt at a converse to this.
Lemma. If $f : A → B$ is a ring homomorphism and $P$ is a prime ideal of $B$, then
$f^{-1}(P)$ is a prime ideal of $A$.
Proof. Notice that $f$ induces a ring homomorphism $g$ from $A$ to $B/P$ by postcomposing with the natural projection map $B → B/P$. Now $a∈\ker g$ if and only if $f(a) ∈ P$, so using the first isomorphism theorem we see that $g$ induces an isomorphism from $A/f^{-1}(P)$ to a subring of $B/P$. Since the latter is an integral
domain, $A/f^{-1}(P)$ must be an integral domain too.
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Czhang271828
发表于 2023-5-31 23:56
