Algebra II - Rings Flashcards

1
Q

Define a ring

A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
2
Q

What are zero divisors

A

If a and b are non-zero elements of a ring R with ab=0 then a and b are called zero divisors

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
3
Q

Define a unit

A

An element a of a ring R is called a unit if it has a two-sided inverse under multiplication. there exists b in R such that ab = ba = 1

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
4
Q

Define a division ring

A

A non-zero ring R is called a division ring if R \ {0} is a group under multiplication - all non zero elements are units

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
5
Q

Define a subring

A

A subset S of a ring R is called a subring of R if it forms a ring under the same operations as R with the same identity element

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
6
Q

Define an isomorphism between two rings

A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
7
Q
A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
8
Q

State the chinese remainder theorem

A

The rings Zm x Zn and Zmn are isomorphic if and only if m and n are coprime

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
9
Q

Prove the chinese remainder theorem

A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
10
Q

Find an integer x with x = 5 mod 8 and x = 6 mod 19

A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
11
Q

Define a ring homomorphism

A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
12
Q

Define an ideal

A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
13
Q

How does addition for cosets work

A

(I + a1) + (I + a2) = I + (a1 + a2)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
14
Q

State the first isomorphism theorem for rings

A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
15
Q

When is an ideal maximal

A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
16
Q
A
17
Q

Define a Euclidean domain

A
18
Q

Theorem: Any Euclidean domain is a principal ideal domain

A
19
Q

What two statements are equivalent to x|y for all x,y in R

A

y is in (x)

(y) is a subset of (x)

20
Q

For x,y in R, when are x and y associates

A

x divides y and y divides x

21
Q

What two statements are equivalent to x is an associate y

A

(y) = (x)

there exists a unit q in R with x = qr

22
Q

Let r be in R \ {0}, when is r irreducible and when is r prime

A
23
Q

Proposition: If R is a domain, then any prime element of R is irreducible

A
24
Q

Proposition: If R is a principal ideal domain, then any irreducible element of R is prime

A
25
Q

When is a complex number algebraic and transcendental

A
26
Q

Define a UFD

A
27
Q

Proposition: If R is a UFD, then every irreducible element of R is prime

A
28
Q

In an integral domain R what is the link between irreducibles in R and R[x]?

A

they’re the same

29
Q

Define a primitive polynomial

A
30
Q

State Gauss’ lemma

A

A primitive irreducible polynomial in Z[x] remains irreducible in Q[x]

31
Q

State Eisenstein’s criterion and prove it

A