Rings

1. Determine whether or not each of the following sets of numbers is a ring under ordinaryaddition and multiplication.

(a) R = the set of positive integers and zero.

(b) R = {kn |n 2 Z, k is a fixed integer}(c) R = {a+ b

p2 | a, b 2 Z}

(d) R = {a+ bp2 + c

p3 | a 2 Z, b, c 2 Q}

2. Define a new multiplication on Z by the rule a�b = 1 for all a, b 2 Z. If + represents ordinaryaddition, determine if (Z,+,�) is a ring.

3. Define the binary operations � and � on Z by

• x� y = x+ y � 7

• x� y = x+ y � 3xy

for all x, y 2 Z. Determine if (Z,�,�) is a ring or not.

4. Consider the set R = {s, t, x, y} endowed with addition and multiplication as defined in thetables below. (R,�,�) is a ring.

� s t x y

s y x s t

t x y t s

x s t x y

y t s y x

� s t x y

s y y x x

t y y x x

x x x x x

y x x x x

(a) What is the zero for this ring?

(b) What is the additive inverse of each element?

(c) What is t(s+ xy)?

(d) Is the ring commutative?

(e) Does the ring have a unity?

(f) Find a pair of zero divisors.

5. Consider the set R = {s, t, v, w, x, y} endowed with addition and multiplication as defined inthe tables below. (R,�,�) is a ring.

� s t v w x y

s s t v w x y

t t v w x y s

v v w x y s t

w w x y s t w

x x y s t v x

y y s t v w y

� s t v w x y

s s s s s s s

t s t v w x y

v s v s x v x

w s w s w s w

x s x v s x v

y s y x w v t

(a) What is the zero for this ring?

(b) What is the additive inverse of each element?

(c) What is t(s+ xy)?

(d) Is the ring commutative?

(e) Does the ring have a unity?

(f) Find a pair of zero divisors.

6. Consider Z12 (the set of integers modulo 12).

(a) Is Z12 a field?

(b) Is Z12 an integral domain?

7. Give an example of a ring that is an integral domain, but is not a field.