[Abstract Algebra] Chapter 4

Question 1

What are the elements in a ring?

Answer 1

a set, an addition operation, a multiplication operation

Question 2

The set and the addition operation in a ring form what type of structure?

Answer 2

an Abelian group

Question 3

What property does the multiplication operation necessarily have?

Answer 3

associativity

Question 4

What property does the multiplication operation most commonly have?

Answer 4

commutativity

Question 5

What type of function is the multiplication operation?

Answer 5

binary function

Question 6

What relation must the multiplication operation have with regards to the addition operation?

Answer 6

multiplication must distribute over addition

Question 7

What is a zero ring?

Answer 7

A ring which consists of only one element

Question 8

Rings (in the Napkin) must contain what two elements?

Answer 8

Additive identity and multiplicative identity

Question 9

What is a product ring?

Answer 9

Ordered pairs (r, s), each element being from a different ring, where operations are done component-wise

Question 10

What is a polynomial ring?

Answer 10

Sets of polynomials with all coefficients in ring R

Question 11

How is the notation for a polynomial ring written?

Answer 11

R[x]

Question 12

How is the notation for a polynomial ring pronounced?

Answer 12

R adjoin x

Question 13

How many variables can a polynomial ring have?

Answer 13

Any number, including infinite

Question 14

What is a unit?

Answer 14

Elements which have a multiplicative inverse

Question 15

What is a field?

Answer 15

Non-trivial rings which all non-zero elements are units

Question 16

How can a zero ring be identified using identity elements?

Answer 16

Additive and multiplicative identities are the same element

Question 17

When is the ring Z/nZ a field?

Answer 17

When n is prime

Question 18

What commonly used sets are fields?

Answer 18

Q, R, C

Question 19

What criteria do ring homomorphisms must have in regards to addition?

Answer 19

f(r1) + f(r2) = f(r1 + r2)

Question 20

What criteria do ring homomorphisms must have in regards to multiplication?

Answer 20

f(r1) * f(r2) = f(r1 * r2)

Question 21

What criteria do ring homomorphisms must have in regards to identities?

Answer 21

f(1R) = 1S, where 1R is the multiplicative identity of R, etc.

Question 22

What is a kernel of a ring homomorphism?

Answer 22

The set of elements which are mapped to 0S under the ring homomorphism

Question 23

What two properties does the kernel of a ring homomorphism have?

Answer 23

Closed under addition, absorbs multiplication from any other element of R

Question 24

What are sets satisfying the two properties of a kernel called?

Answer 24

Ideals

Question 25

What is a proper ideal?

Answer 25

Ideals which are not equal to the ring it is part of

Question 26

How can improper ideals be identified?

Answer 26

They contain a unit

Question 27

How many ideals does a field contain?

Answer 27

Two; the zero ring and the entire field

Question 28

How is the ideal generated by a set of elements, x_i, denoted?

Answer 28

(x_1, x_2, …, x_i, …, x_n)

Question 29

What are ideals containing only one element called?

Answer 29

Principal ideals

Question 30

What is a quotient ring?

Answer 30

{r + I| r in R} for an ideal I of R

Question 31

What type of structure are the elements of a quotient ring?

Answer 31

Equivalence classes

Question 32

How is a quotient ring denoted?

Answer 32

R/I

Question 33

How is a quotient ring pronounced?

Answer 33

R mod I

Question 34

What is a simple way to find elements of a quotient ring?

Answer 34

Assume any element of I is equal to zero, or “modding out” by elements of I

Question 35

What is a principal ideal ring?

Answer 35

A ring where all ideals are principal ideals

Question 36

For a field k, what ring related to k is always a principal ideal ring?

Answer 36

k[x]

Question 37

Is the ideal (10, 15) principal?

Answer 37

Yes, as (10, 15) = 5

Question 38

Why are brackets used to denote the greatest common factor between some amount of numbers?

Answer 38

As the ideal generated by those numbers is congruent to the principal ideal generated by their greatest common factor

Question 39

Is the ideal (x, 2015) principal?

Answer 39

No, as any f(x) generating the ring must divide both x and 2015 which is impossible

Question 40

What is a Noetherian ring?

Answer 40

Rings where every ideal can be generated with finitely many elements

Question 41

What is an example of a non-Noetherian ring?

Answer 41

Z[x_1, x_2, x_3, …] has the ideal (x_1, x_2, x_3, …)

Question 42

What condition regarding chains of ideals do only non-Noetherian rings not satisfy?

Answer 42

The existence of an infinite ascending chain of ideals I1 being a subset of I2, being a subset of I3, …

Question 43

Given a Noetherian ring R, what related ring is always Noetherian?

Answer 43

R[x]