Rings PRACTICE

Question 1

nZ with the usual addition and multiplication

Answer 1

Yes, nZ for n∈Z+ is a commutative ring, but without unity unless n = 1, and is not a field.

Question 2

Z+with the usual addition and multiplication

Answer 2

No, Z+ is not a ring; there is no identity for addition.

Question 3

Z × Z with addition and multiplication by components

Answer 3

Yes, Z×Z is a commutative ring with unit (1, 1), but it is not a field because (2, 0) has no multiplicative inverse.

Question 4

2Z × Z with addition and multiplication by components

Answer 4

Yes, 2Z×Z is a commutative ring, but without unity, and is not a field.

Question 5

{a + b√2 | a, b ∈ Z} with the usual addition and multiplication

Answer 5

Yes, {a+ b√2|a,b∈Z} is a commutative ring with unity, but it is not a field because
2 has no multiplicative inverse.

Question 6

Every field is also a ring.

Answer 6

T

Question 7

Every ring with unity has at least two units.

Answer 7

F

Question 8

Every ring has a multiplicative identity.

Answer 8

F

Question 9

Every ring with unity has at most two units.

Answer 9

F

Question 10

It is possible for a subset of some field to be a ring but not a subfield, under the induced operations.

Answer 10

T

Question 11

The distributive laws for a ring are not very important.

Answer 11

F

Question 12

Multiplication in a field is commutative.

Answer 12

T

Question 13

The nonzero elements of a field form a group under the multiplication in the field.

Answer 13

T

Question 14

The addition in every ring is commutative.

Answer 14

T

Question 15

Every element in a ring has an additive inverse.

Answer 15

T

Question 16

The concept of a ring homomorphism is closely connected with the idea of a factor ring.

Answer 16

T

Question 17

A ring homomorphism φ : R → R carries ideals of R into ideals of R

Answer 17

F

Question 18

A ring homomorphism is one-to-one if and only if the kernel is {0}.

Answer 18

T

Question 19

Q is an ideal in R.

Answer 19

F

Question 20

Every ideal in a ring is a subring of the ring.

Answer 20

T

Question 21

Every subring of every ring is an ideal of the ring.

Answer 21

F

Question 22

Every quotient ring of every commutative ring is again a commutative ring.

Answer 22

T

Question 23

The rings Z/4Z and Z4 are isomorphic.

Answer 23

T

Question 24

An ideal N in a ring R with unity 1 is all of R if and only if 1∈N.

Answer 24

T

Question 25

The concept of an ideal is to the concept of a ring as the concept of a normal subgroup is to the concept of
a group.

Answer 25

T