[Abstract Algebra] Chapter 5

Question 1

What are zero divisors?

Answer 1

Two nonzero elements of R which multiply to zero

Question 2

What are examples of zero divisors in the ring Z/15Z?

Answer 2

3, 5

Question 3

What is a ring without any zero divisors called?

Answer 3

Integral domains

Question 4

What special property do fields have in relation to integral domains?

Answer 4

Fields are always integral domains

Question 5

What special property do fields have in relation to integral domains?

Answer 5

Fields are always integral domains

Question 6

What is a PID?

Answer 6

An integral domain with all ideals principal (Principal Ideal Domain)

Question 7

What is a prime ideal?

Answer 7

A proper ideal I such that any element xy in I has either x in I or y in I

Question 8

What is an example of a non-prime ideal in Z?

Answer 8

(8)

Question 9

How can a prime ideal be identified using quotient groups?

Answer 9

I is a prime ideal iff R/I is an integral domain

Question 10

What is a maximal ideal?

Answer 10

A proper ideal I such that the ideal is not contained in any other proper ideal

Question 11

What is the relationship between prime and maximal ideals?

Answer 11

Maximal ideals are always prime

Question 12

What is an example of a maximal ideal in Z[x]?

Answer 12

(x, 5)

Question 13

How can a maximal ideal be identified using quotient groups?

Answer 13

I is a maximal ideal iff R/I is a field

Question 14

What types of rings have a field of fractions?

Answer 14

Integral domains

Question 15

What elements does a field of fractions contain?

Answer 15

Elements a/b such that both a and b are members of R, and b is nonzero

Question 16

What are the equivalence classes in a field of fractions?

Answer 16

a/b ~ c/d iff bc = ad

Question 17

What are the two operations in a field of fractions?

Answer 17

a/b * c/d = ac/bd, a/b + c/d = (ad+bc)/bd

Question 18

How is a field of fractions denoted?

Answer 18

Frac(k)

Question 19

What is a field of rational functions?

Answer 19

Frac(k[x])

Question 20

How is a field of rational functions denoted?

Answer 20

Frac(x)

Question 21

What is an irreducible element?

Answer 21

An element which cannot be written as the product of two non-units

Question 22

What is an UFD?

Answer 22

An integral domain where every nonzero non-unit element can be written as the product of irreducible elements, unique up to multiplication by units (Unique Factorization Domains)

Question 23

What is a degenerate example of a UFD?

Answer 23

A field, as there are no nonzero non-unit elements

Question 24

What is the relationship between UFDs and PIDs?

Answer 24

Every PID is a UFD

Question 25

What related UFD exists given an UFD R?

Answer 25

R[x]