Ideals

Question 1

Ideal

Answer 1

A subring of A of a ring R is called a (two-sided) ideal of R if for every r∈R and every a∈A both ra and ar are in A

Question 2

Ideal test

Answer 2

A nonempty subset A of a ring R is an ideal of R if
1. a-b∈A whenever a,b ∈ A (closed under addition)
2. r.a and a.r are in A, and whenever a∈A and r∈R (absorbed under multiplication by any ring element).)

Question 3

Existence of Factor Rings

Answer 3

Let R be a ring and let A be a subring of R.
The set of cosets {r+A|r∈R} is a ring under the operations
(s + A)+(t + A) = s + t + A and (s + A)(t + A) =st + A if and only if A is an ideal of R.

Question 4

Prime Ideal

Answer 4

A prime ideal A of a commutative ring R is a proper ideal of R such that a, b ∈ R and ab ∈ A imply a ∈ A or b ∈ A.

In other words, the product of two element belongs to an ideal implies one of them must be in I

Question 5

Maximal Ideals

Answer 5

A maximal ideal of a commutative ring R is a proper ideal of R such that, whenever B is an ideal of R and A ⊆ B ⊆ R, then B = A or B = R.

A maximal ideal is an ideal in a ring that is as large as it can be without becoming the whole ring. In other words, it is a “boundary” ideal, where if you try to make it any bigger by adding more elements, it will become the entire ring.

Question 6

R/A Is an Integral Domain If and Only If A Is Prime

Answer 6

Let R be a commutative ring with unity and let A be an ideal of R. Then R/A is an integral domain if and only if A is prime.

Question 7

R/A Is a Field If and Only If A Is Maximal

Answer 7

Let R be a commutative ring with unity and let A be an ideal of R. Then R/A is a field if and only if A is maximal.

Question 8

Kernels Are Ideals

Answer 8

Let ϕ be a ring homomorphism from a ring R to a ring S. Then Kerϕ = {r∈R |ϕ (r) = 0} is an ideal of R.

Question 9

Principal Ideals

Answer 9

Let R be any commutative ring and a ∈ R

⟨a⟩ = {b ∈ R : b = ax, for some x ∈ R}.
We then call ⟨a⟩ a principal ideal and call a a generator for ⟨a⟩.

Question 10

All Ideals in Z Are Principal

Answer 10

All Ideals in Z Are Principal

Question 11

The finite integral domain is a field therefore Maximal and Prime ideal coincides

Answer 11

The finite integral domain is a field therefore Maximal and Prime ideal coincides􏰆􏰂􏰃 􏰋􏰈􏰎􏰖􏰃 􏰄􏰉􏰊 􏰆􏰂􏰃 􏰖􏰄􏰚􏰎􏰖􏰄􏰍 􏰎􏰊􏰃􏰄􏰍􏰏 􏰕􏰇􏰎􏰉􏰕􏰎􏰊􏰃􏰔

Question 12

A maximal ideal of a ring R is an ideal M such that there is no ideal N of R such that M⊂N⊂R

Answer 12

A maximal ideal of a ring R is an ideal M such that there is no ideal N of R such that M⊂N⊂R

Question 13

A principal ideal of a commutative ring with unity is an ideal N with the property that there exists a ∈ N such that N is the smallest ideal that contains a.

Answer 13

A principal ideal of a commutative ring with unity is an ideal N with the property that there exists a ∈ N such that N is the smallest ideal that contains a.

Question 14

A prime field is a field that has no proper subfields.

Answer 14

A prime field is a field that has no proper subfields.

Question 15

Prop: Letφ: R==>S be a homomorphism I= kerφ
then ‘‘I’’ has to satisfy these 4 conditions

Answer 15

1. 0∈I
2. ∀f,g∈I==>f+g ∈I
3. ∀f∈I==>-f∈I
4. ∀r∈R and ∀f∈I==>r.f ∈I

Question 16

A subset satisfies
1. 0∈I
2. ∀f,g∈I==>f+g ∈I
3. ∀f∈I==>-f∈I
4. ∀r∈R and ∀f∈I==>r.f ∈I

Answer 16

is an ideal