September

Question 1

Binary Operation

Answer 1

Let X be a binary operation on X is a map
X*X—-»X
(x,y)==>x.y

Question 2

Properties

Answer 2

For all x,y,z ∈ X

1. Associative (x.y).z=x.(y.z)
2. Commutative x.y=y.x
   3 Admit neutral iff x.e=e.x=x
3. Admit the inverses x.y=y.x=e

Question 3

Group

Answer 3

is a set G with BO * that is assoc, commu, neutral and inverses

Abelian or commutative

Question 4

Let (G,*) be a group with neutral elements e

Answer 4

IF
x.y=e and x.z=e then y=z
y.x=e and z.x=e then y=z

Question 5

Definition of RIng

Answer 5

Ring is a set with 2 BO +, *

Commutative, associative, it has additive identity 0, inverse is -a+a=0, distributive over multiplication

A ring with 1 or a ring with unity, if ring has a neutral elements

Question 6

A subring

Answer 6

A subring S of a ring (R,+,) is a subset S⊆R such that +, restricted to BO on S

A subset S of a ring R is a subring of R if S is itself a ring with the operations of R.

Question 7

Subring test

Answer 7

A non-empty subset S of R is a subring if S is closed under add and multi

if a-b and ab is closed under S whenever a and b are in S

Question 8

What is Q?

Answer 8

Pairs of integers (a,b) b/=0

1/2 (1,2)
2/4 (2,4)

Question 9

Equivalence relations

Answer 9

1.Reflexivity ∀a∈R, aRa (a,a)∈R
2. Symmetry ∀a,b∈R aRb<===>bRa
3. ∀a,b,c ∈R if aRb and bRc==> aRc

Question 10

Zero Divisor

Answer 10

A zero divisor is a non-zero element a of the commutative ring R such that there is a non-zero element b∈R with a.b=0

Question 11

Integral domain

Answer 11

is a commutative ring with unity and no zero-divisors

Question 12

r∈R is a unit if

Answer 12

there some s such that r.s=1

and r has a mul inverse

{r∈R|r is a unit}

Question 13

A field i

Answer 13

is a commutative ring with unity in which every non-zero element is a unit

Question 14

A finite integral domain is field

Answer 14

A finite integral domain is field

Question 15

Zp is a field

Answer 15

Zp is a field

Question 16

Characteristic of integral domain

Answer 16

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

Question 17

Characteristic of rings

Answer 17

If for a ring R a positive integer n exists such that n·a=0 for all a∈R, then the least such positive integer is the characteristic of the ring R. If no such positive integer exists, then R is of characteristic 0.

Question 18

Characteristic of a field

Answer 18

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 19

Mapping

Answer 19

1. Injective (aka 1-1) if ∀ a,b ∈X, f(a)=f(b) ==>a=b
2. Surjective (aka onto) if ∀ y∈Y, and ∃x∈X such that f(x)=y, (If each element of Y is the image of at least one element of X)
3. Bijective meaning both 1 and 2

Question 20

Connections

Answer 20

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 21

Definitions Ring Homomorphism

Answer 21

A ring homomorphism φ from a ring R to a ring S is a mapping from Rto S that preserves the two ring operations; that is, for all a, b in R
φ(a+b)=φ(a)+φ(b)
φ(ab)=φ(a)φ(b)
φ(1R)=1S

Question 22

Ring Isomorphism

Answer 22

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

Question 23

Cancellation

Answer 23

Let a, b, and c belong to an integral domain. If a not = 0 and ab=ac, then b - c.

PROOF From ab = ac, we have a(b-c) = 0. Since a not= 0, we must have b-c = 0.

Question 24

φ is injective <==>ker(φ)=0

Question 25

Ring Isomorphism

Answer 25

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

Let homo R==>S the mapping from R/Kerφ to φ(R)is given by r+kerφ==>φ(r), is an isomorphism in symbols R/kerφ=φ(R)