Rings

Question 1

A binary operation

Answer 1

Let A be a set. A binary operation on A is a function * A x A -> A

Question 2

Ring

Answer 2

A ring is a set R along with 
- a binary operation + on R called addition
- a binary operation . on R called multiplication
satisfying the following axioms
- closure under addition
- associativity of addition
- commutativity of addition
- existence of an additive inverse
- existence of a additive identity
- closure under multiplication 
- associativity of multiplication
- left distributivity
- right distributivity

Question 3

Ring with one

Answer 3

• existence of a multiplicative identity

Question 4

Commutative ring

Answer 4

• commutativity of multiplication

Question 5

Zero divisor

Answer 5

Let R be a commutative ring and let a e R. Then a is a zero divisor if there exists b e R s.t. b=/0 and ab=0

Question 6

Integral domain

Answer 6

Let R be a commutative ring with one. We say R is an integral domain if there are no zero divisors in R

Question 7

Unit

Answer 7

Let R be a ring with one. Let a e R. We say that a is a unit if there exists a^(-1) e R s.t aa^(1) = 1 = a^(-1)a.

Question 8

Set of units

Answer 8

Let U(R) = { a e R : a is a unit}

Question 9

Field

Answer 9

Let R be a commutative ring with one. The R is a field if the existence of a multiplicative inverse holds.

Question 10

Unit/ Zero Divisor

Answer 10

Let R be a commutative ring with one and let a e R with a=/0. Suppose a is a unit then a is not a zero divisor.

Question 11

Field/ Integral domain

Answer 11

Let F be a field. Then F is an integral domain.

Question 12

Set of polynomials over F (F[X])

Answer 12

The set of all polynomials over F.

Question 13

Element of F[X]

Answer 13

expression of the form f(X) = anX^n + …. + a0

Question 14

The ring of polynomials over F

Answer 14

The set of F[x] with addition and mulplication

Question 15

How degree interacts with addition and multiplication

Answer 15

• f(x) + g(x) = 0 or deg( f(x) + g(x) ) <= max {deg(f(x) , deg(g(x)}
• f(x)g(x) =/0 and deg {f(x)g(x)} = deg(f(x) + deg(g(x)

Question 16

Set of polynomials over R (R[X])

Answer 16

Let R be a ring and X be an indeterminate. This is defined in the same way as a polynomial over F however, we allow coefficients from R

Question 17

Subring

Answer 17

Let R be a ring and let S be a subset of R. We say S is a subring of R if it is a ring with addition and multiplication from R.

Question 18

First subring test

Answer 18

• 0 e S

- for all x, y e S, we have -x e S, x+y e S and xy e S

Question 19

Second subring test

Answer 19

• 0 e s

- for all x, y e S, we have x-y e S and xy e S