[Abstract Algebra] Chapter 5 Flashcards
What are zero divisors?
Two nonzero elements of R which multiply to zero
What are examples of zero divisors in the ring Z/15Z?
3, 5
What is a ring without any zero divisors called?
Integral domains
What special property do fields have in relation to integral domains?
Fields are always integral domains
What special property do fields have in relation to integral domains?
Fields are always integral domains
What is a PID?
An integral domain with all ideals principal (Principal Ideal Domain)
What is a prime ideal?
A proper ideal I such that any element xy in I has either x in I or y in I
What is an example of a non-prime ideal in Z?
(8)
How can a prime ideal be identified using quotient groups?
I is a prime ideal iff R/I is an integral domain
What is a maximal ideal?
A proper ideal I such that the ideal is not contained in any other proper ideal
What is the relationship between prime and maximal ideals?
Maximal ideals are always prime
What is an example of a maximal ideal in Z[x]?
(x, 5)
How can a maximal ideal be identified using quotient groups?
I is a maximal ideal iff R/I is a field
What types of rings have a field of fractions?
Integral domains
What elements does a field of fractions contain?
Elements a/b such that both a and b are members of R, and b is nonzero
What are the equivalence classes in a field of fractions?
a/b ~ c/d iff bc = ad
What are the two operations in a field of fractions?
a/b * c/d = ac/bd, a/b + c/d = (ad+bc)/bd
How is a field of fractions denoted?
Frac(k)
What is a field of rational functions?
Frac(k[x])
How is a field of rational functions denoted?
Frac(x)
What is an irreducible element?
An element which cannot be written as the product of two non-units
What is an UFD?
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)
What is a degenerate example of a UFD?
A field, as there are no nonzero non-unit elements
What is the relationship between UFDs and PIDs?
Every PID is a UFD
What related UFD exists given an UFD R?
R[x]
