Modern Algebra Definitions

Front 1

Homomorphism

Back 1

Let R and S be rings. A function h:R-S is called a ___ if the following 2 properties hold:

1. For all x,y in R, h(x+y)=h(x)+h(y)
-h preserves addition
2. For all x,y in R, h(xy)=h(x)*h(y)
-h preserves multiplication

Front 2

Characteristic

Back 2

If there exists a positive integer n such that nx=0r for each element x in a ring R, the smallest such positive integer n is called the ____ of R.

Front 3

Subring

Back 3

Let R be a ring and let S be a subset of R with addition and multiplication as in R. Then S is a ___ of R if and only if 0r is in S, and whenever x,y are in S, then x-y is in S and xy is in S.

Front 4

Integral Domain

Back 4

A commutative ring with identity is called an ____ if it has no zero divisors

Front 5

Isomorphism

Back 5

A homomorphism that is both one-to-one and onto is called an ___.

Front 6

Field

Back 6

A commutative ring with identity is called a ___ if every nonzero element has a multiplicative inverse.

Front 7

Zero Divisor

Back 7

A nonzero element z in a ring is called a ___ if there exists a nonzero element r in R such that rz=0 and zr=0.

Front 8

Well Defined

Back 8

If x = x' , then h(x) = h(x' )

Front 9

One-to-One

Back 9

If h(x) = h(y), then x = y

Front 10

Onto

Back 10

For all y in S, there exists an x in R such that h(x) = y

Front 11

Reflexive (equivalence relation)

Back 11

a is related to a

Front 12

Symmetric (equivalence relation)

Back 12

if a is related to b, then b is related to a.

Front 13

Transitive (equivalence relation)

Back 13

If a is related to b and b is related to c, then a is related to c.

Front 14

α = a + bi...what are a and b in terms of r and θ?

Back 14

a = r*cos(θ)

b = r*sin(θ)

Front 15

α = a + bi...what is r and θ?

Back 15

r = √(a^2+b^2 )

θ = tan^(-1)〖b/a〗

Front 16

what is the conjugate of α? α bar?

Back 16

/α = a - bi

Front 17

what is the length of α? |α|

Back 17

|α| = √(a^2+b^2 )

Front 18

(a+bi)+(c+di) =...

Back 18

(a+c)+(b+d)i

Front 19

(a+bi)*(c+di) =...

Back 19

(ac-bd)+(ad+bc)i

Front 20

Trigonometric Form of α

Back 20

α = r*cos(θ) + i*r*sin(θ)

Front 21

α^n = ...

Back 21

= r^n [cos(nθ)+i*sin(nθ)]...taken nθ congruent mod 360

Front 22

n√(α) = ...

Back 22

= n√(r)*[cos(θ+360k/n)+i*sin(θ+360k/n)]

k = 0, 1, 2, ..., n-1

Front 23

Chinese Remainder Theorem

Back 23

x=a(mod m)
x=b(mod n)

x=a+mu(b-a) (mod mn)

Front 24

Multiplicative Inverse

Back 24

Suppose R is a ring with identity and let x be in R. If there exists an element y in R such that xy=yx=1, then y is called a ___ for x.

Front 25

Ring

Back 25

A nonempty set R together with two operations called addition and multiplication that satisfy 7 axioms:

1. For all x,y in R, x+y is in R and xy is in R. (R is closed under addition and multiplication)
2. For all x,y in R, x+y = y+x (addition is commutative)
3. For all x,y,z in R, x+(y+z) = (x+y)+z (addition is associative)
4. There is an element 0r in R such that x+0r = x for every x in R. (R contains a zero element)
5. For each x in R, there exists an element y in R such that x+y=0r. (Each element of R has an additive inverse)
6. For all x,y,z in R, x(yz)=(xy)z. (Multiplication is Associative)
7. For all x,y,z in R, x(y+z)=xy+xz and (y+z)x=yx+zx (The Distributive Properties hold in R).

Front 26

Extra Ring Axioms

Back 26

8. For all x,y in R, xy=yx (multiplication is associative)
9. There is some nonzero element 1r in R such that 1r*x = x*1r = x for every x in R. (R contains an identity element).