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An additive Subgroup
of a ring is called an Ideal if it satisfies -
This is analogous to the formulation for normal groups in the sense that ideals can be used to construct factor rings.
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Intuition: Ideals generalize the notion of sets of multiples. In particular, consider how any integer
multiplied to a multiple of is also a multiple of . In the same vein, any ring element
multiplied (whether right or left) to an element of the ideal is also an element of the ideal.
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Every nonzero ring
has at least two ideals — the improper ideal and the trivial ideal -
(Fraleigh e26.22) Let
be an ideal of and a Ring Homomorphism. Then is an ideal of although it need not be an ideal of Also if
is an ideal of either or , then is also an ideal of . -
(Fraleigh e26.27) The intersection of ideals of ring
is also an ideal of . -
A maximal ideal of a ring
is an ideal such that there is no proper ideal of properly containing . -
(Fraleigh 27.9) Let
be a Commutative Ring with unity. Then is a maximal ideal of if and only if is a field -
Proof: In the forward direction, any additive coset
where must have a multiplicative inverse. Otherwise we can construct a non-proper ideal containing the maximal ideal . In the converse direction, if
is a field, Let be an ideal, then if , we have that using the canonical homomorphism is an ideal of . It contains the zero ideal which is a contradiction since fields only have two ideals.
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An ideal
in commutative ring is a prime ideal if -
The motivation behind this definition is that the factor ring
will be an Integral Domain if and only if is a prime ideal. That is
has no divisors of . See more in (Fraleigh 27.15) -
Intuition: Another way to look at this is as a generalization of Euclid’s Lemma for primes. We can state is as follows
If
is a part of the prime ideal (i.e., is “prime”), then either is in the prime ideal or is in the prime ideal (i.e., either or are “prime”).
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(Fraleigh 27.16) Every maximal ideal in a commutative ring
with unity is a prime ideal. - Proof: If
is a maximal ideal, then is a field, which is an integral domain and therefore maximal. - Intuition: A concrete example of this is found in
. Consider where is a prime. It is easy to verify that it is an ideal. However, it is not maximal. In fact is maximal. It contains all multiples of prime and clearly for all .. It is also a prime ideal.
- Proof: If
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(Fraleigh e27.24) For a finite commutative ring with unity
, every prime ideal is a maximal ideal. -
If
is a commutative ring with unity and , the ideal of all multiples of is called the principal ideal generated by defined as An ideal is called a principal ideal if
for some . That is it is an ideal generated by exactly one element - Intuition: We can think of a principal ideal
as the set of “multiples” of - Every ideal of
is a principal ideal. - If we have
then we denote the ideal generated by these as
- Intuition: We can think of a principal ideal