Chapter 3

Question 1

RING

Answer 1

A ring is an algebraic structure that consists of a nonempty set R and two binary operations on R, called addition and multiplication and denoted “+” and “·”, such that the following properties (ring axioms) hold for all a, b, c ∈ R.

Question 2

8 AXIOMS

Answer 2

(1) If a ∈ R and b ∈ R, then a + b ∈R.
(2) a + (b + c) = (a + b) + c.
(3) a + b = b + a.
(4) ∃0R∈R ∀a∈R a + 0R = a = 0R + a.
(5) For each a ∈ R, the equation
a + x = 0R has a solution in R.
(6) If a ∈ R and b ∈ R, then ab ∈ R.
(7) a(bc) = (ab)c.
(8) a(b + c) = ab + ac, (a + b)c = ac + bc.

Question 3

COMMUTATIVE RING

Answer 3

A commutative ring is a ring R such that

(9) ∀a,b∈R ab = ba.

Question 4

RING WITH IDENTITY

Answer 4

A ring with identity is a ring R that contains an element 1R satisfying
(10) ∀a∈R a · 1R = a = 1R · a.

Question 5

INTEGRAL DOMAIN

Answer 5

An integral domain is a commutative ring R with identity 1R /= 0R and such that
(11) ∀a,b∈R ab = 0R ⇒ a = 0R or b = 0R.

Question 6

FIELD

Answer 6

A field is a commutative ring R with identity 1R /= 0R and such that
(12) ∀a∈R a /= 0R ⇒ ax = 1R has a solution in R.

Question 7

PRODUCT RING

Answer 7

If R and S are rings, then the Cartesian product R ×S is also a ring, called the product ring, with the binary operations defined as follows…
(r, s) + (r!, s!) = (r + r!, s + s!) and
(r, s) · (r!, s!) = (rr!, ss!).

Question 8

SUBRING

Answer 8

If R is a ring and S is a nonempty subset of R that is itself a ring under the addition and multiplication in R, then S is a subring of R.

Question 9

PROPERTIES OF A SUBRING

Answer 9

Theorem. Let R be a ring and S a subset of R such that
(i) ∀a,b∈S a + b ∈ S,
(ii) ∀a,b∈S ab ∈ S,
(iii) 0R ∈ S,
(iv) ∀a∈S a + x = 0R has a solution in S.
Then S is a subring of R.

Question 10

ADDITIVE INVERSE

Answer 10

The unique solution of
a + x = 0R in R is called the additive inverse of a and is denoted by −a. Thus
a + (−a) = 0R = (−a) + a.

Question 11

SUBTRACTION

Answer 11

For any ring R, the binary operation on R given by

(a, b) → a − b := a + (−b) is called the subtraction in R.

Question 12

MULTIPLICATIVE INVERSE

Answer 12

Let R be a ring with identity and a ∈ R. Then a is called a unit in R if there exists an element u ∈ R such that au = 1R = ua. In this case, u is called the multiplicative inverse of a and is denoted by a−1.

Question 13

ZERO DIVISOR

Answer 13

Let R be a ring and a ∈ R. Then a is a zero divisor in R if

(1) a /= 0R and
(2) there is a nonzero element b ∈ R such that either ab = 0R or ba = 0R.

Question 14

ISOMORPHIC

Answer 14

A ring R is isomorphic to a ring S (written R
∼=S) if there is a function f : R → S such that
(1) f is injective,
(2) f is surjective,
(3) f (a + b) = f (a) + f (b) for all a, b ∈ R,
(4) f (ab) = f (a)f (b) for all a, b ∈ R.
In this case, the function f is called an isomorphism.

Question 15

HOMOMORPHISM

Answer 15

Let R and S be rings. A function f : R → S is a homomorphism if
f (a + b) = f (a) + f (b) and
f (ab) = f (a)f (b) for all a, b ∈ R.