Rings Concepts

Question 1

Function mapping X—->Y

Answer 1

X is domain and is Codomain

Question 2

Injective

Answer 2

If for all a,b ∈X, f(a)=f(b), implies a=b

Question 3

Surjective

Answer 3

Onto function
If for all y∈Y, and for some x∈X f(x)=y
(If each element of Y is the image of at least one element of X)

Question 4

Bijective

Answer 4

Both one to one and onto
For all y∈Y, there’s exactly one x∈X f(x)=y

Question 5

Connections

Answer 5

1. If it’s surjective iff f(x)=y
2. If it’s injective, if for all y∈Y f^-1(y)has at most one element
3. If it’s bijective <====> for all y∈Y f^-1(y) has exactly one element

Question 6

If f is surjective

Answer 6

F(x)=y <===>for all y∈Y, for some x∈X | x∈f^-1(y)
F(x)=y <===>for all y∈Y f^-1(y) is non-empty

Question 7

If p is prime, then Z/pZ is an integral domain

Answer 7

If p is prime, then Z/pZ is an integral domain

Question 8

Ring Isomorphism

Answer 8

A ring homomorphism that is both one-to-one and onto is called a ringisomorphism.

Question 9

Properties of Homomorphisms #1

Answer 9

Let ϕ be a ring homomorphism from a ring R to a ring S. Let A be a subring of R and let B be an ideal of S.

1. For any r ∈ R and any positive integer n,
   ϕ(nr) =nϕ(r) and (ϕ(r))^n
2. ϕ(A) = {ϕ(a) | a ∈ A} is a subring of S.
3. If A is an ideal and ϕ is onto S, then ϕ(A) is an ideal.
   4.ϕ^-1(B) = {r ∈ R|ϕ(r) ∈ B} is an ideal of R.

Question 10

An isomorphism of a ring R with a ring R′ is a homomorphism φ:
R →R′mapping R onto R′ such that Ker(φ )={0}.

Answer 10

An isomorphism of a ring R with a ring R′ is a homomorphism φ:
R →R′mapping R onto R′ such that Ker(φ )={0}.

Question 11

Properties of Homomorphisms #2

Answer 11

Let ϕ be a ring homomorphism from a ring R to a ring S. Let A be a subring of R and let B be an ideal of S.

5. If R is commutative, then ϕ(R) is commutative.
6. If R has a unity 1, S /= {0}, and ϕ is onto,
   then ϕ(1) is the unity of S.
7. ϕ is an isomorphism if and only if ϕ is onto and Ker ϕ ={r ∈ R | ϕ(r) = 0} = {0}.
8. If ϕ is an isomorphism from R onto S, then ϕ^-1 is an isomorphism from S onto R.

Question 12

The kernel of a homomorphism φ mapping a ring R into a ring R′ is
{ r∈R|φ(r)=0’ }.

Answer 12

The kernel of a homomorphism φ mapping a ring R into a ring R′ is
{ r∈R|φ(r)=0’ }.

Question 13

What is the field of quotients of an integral domain?

Answer 13

The field of quotients (or field of fractions) of an integral domain D is the set of all elements of the form a/b and a,b∈D, and 𝑏≠0. This set forms a field 𝐹 where:
add: a/b+c/d=(ad+bc)/bd
mul: a/b*c/d=ac/bd D is embedded in F by identifying each a∈D with a/1. This allows division by any non-zero element of D, forming a field

Question 14

What is the characteristic of a ring?

Answer 14

For a ring R, the characteristic is the smallest positive integer n such that n.a=0 for all a∈R where n.a means a adding itself n times. if no n exists, the ring has characteristic 0.

Question 15

What is the characteristic of a field?

Answer 15

A field has a characteristic of either 0 or a prime p. If the characteristic is 0, no integer multiple of 1 equals 0. If the characteristic is p, then 𝑝⋅1=0 for all elements in the field.

Question 16

What is the characteristic of an integral domain?

Answer 16

The characteristic of an integral domain is either 0 or a prime number p. If the characteristic were composite, it would introduce zero divisors, which contradicts the definition of an integral domain.

Question 17

φ is injective <=====>ker(φ)={0}=0

Answer 17

φ is injective <=====>ker(φ)={0}=0

Question 18

An isomorphism of a ring R with a ring R′ is a homomorphism φ : R → R′, mapping R onto R’ such that Ker(φ) = {0}.

Answer 18

An isomorphism of a ring R with a ring R′ is a homomorphism φ : R → R′, mapping R onto R’ such that Ker(φ) = {0}.

Question 19

An ideal N of a ring R is an additive subgroup of ⟨R,+⟩such that for all r∈R and all n∈N, we have rn∈N and nr ∈ N.

Answer 19

An ideal N of a ring R is an additive subgroup of ⟨R,+⟩such that for all r∈R and all n∈N, we have rn∈N and nr ∈ N.

Question 20

The kernel of a homomorphism φ mapping a ring R into a ring R′ is
{r ∈ R|φ(r) = 0′}.

Answer 20

The kernel of a homomorphism φ mapping a ring R into a ring R′ is
{r ∈ R|φ(r) = 0′}.