Chapter 3: Localisation

These flashcards cover key concepts, definitions, and theorems related to the localization and the construction of rings of fractions as discussed in Chapter 3.

What is the key principle behind the construction of the ring of fractions in relation to Z and Q?

The construction models the relationship by inverting certain elements of Z, leading to the smallest ring containing Z where every nonzero element is invertible.

What is the equivalence relation defined on the set Z × Z×?

(a, b) ≡ (c, d) if and only if ad = bc.

How are equivalence classes of (Z × Z×)/ ≡ related to Q?

The equivalence classes are in one-to-one correspondence with Q.

Provide the operations for addition and multiplication of equivalence classes in S−1R.

[(a, b)] + [(c, d)] = [(ad + bc, bd)], [(a, b)] · [(c, d)] = [(ac, bd)].

What are the axioms that define the subset S in a commutative unital ring R?

(Q1) 0 ̸∈ S, (Q2) S does not contain any zero divisor, (Q3) S is a multiplicative set.

What does the theorem established in 3.4 state about the ring S−1R?

S−1R is a commutative unital ring with specific well-defined operations, and it is the smallest commutative unital ring containing R where elements of S are invertible.

What is the significance of the statement 'if R is an integral domain, then S−1R is also an integral domain'?

This states that the property of being an integral domain is preserved under the localization process.

What is the field of fractions of the ring of integers Z?

The field of fractions of Z is Q (the rational numbers).

What does the definition of the localization of an integral domain at a prime ideal imply?

It creates a ring of fractions that allows inverses for elements not in the prime ideal.

What contradiction arises from assuming a proper ideal of Rp contains the element r s where r ∈ S?

It leads to the conclusion that 1 must be in the ideal, which is a contradiction because it indicates the ideal cannot be proper.

What is the key principle behind the construction of the ring of fractions in relation to Z and Q?

The construction models the relationship by inverting certain elements of Z, leading to the smallest ring containing Z where every nonzero element is invertible.

What is the equivalence relation defined on the set Z × Z×?

(a, b) ≡ (c, d) if and only if ad = bc.

How are equivalence classes of (Z × Z×)/ ≡ related to Q?

The equivalence classes are in one-to-one correspondence with Q.

Provide the operations for addition and multiplication of equivalence classes in S−1R.

[(a, b)] + [(c, d)] = [(ad + bc, bd)], [(a, b)] · [(c, d)] = [(ac, bd)].

What are the axioms that define the subset S in a commutative unital ring R?

(Q1) 0 ̸∈ S, (Q2) S does not contain any zero divisor, (Q3) S is a multiplicative set.

What does the theorem established in 3.4 state about the ring S−1R?

S−1R is a commutative unital ring with specific well-defined operations, and it is the smallest commutative unital ring containing R where elements of S are invertible.

What is the significance of the statement 'if R is an integral domain, then S−1R is also an integral domain'?

This states that the property of being an integral domain is preserved under the localization process.

What is the field of fractions of the ring of integers Z?

The field of fractions of Z is Q (the rational numbers).

What does the definition of the localization of an integral domain at a prime ideal imply?

It creates a ring of fractions that allows inverses for elements not in the prime ideal.

What contradiction arises from assuming a proper ideal of Rp contains the element r s where r ∈ S?

It leads to the conclusion that 1 must be in the ideal, which is a contradiction because it indicates the ideal cannot be proper.

What is a multiplicative set S in a ring R?

A subset S of R is a multiplicative set if 1 \in S and for any s1, s2 \in S, their product s1s2 \in S.

What is a zero divisor in a ring?

An element a \neq 0 in a ring R is a zero divisor if there exists a nonzero element b \in R such that ab = 0.

What does Z^\times specifically denote in the context of the equivalence relation for constructing Q from Z?

Z^\times denotes the set of all non-zero integers, which are the elements allowed in the denominator of the rational numbers.