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Magazine Problem 8
Decide whether the indicated operations of addition and multiplication are defined (closed) on the set, and give a ring structure. If a ring is not formed, tell why this is the case. If a ring is formed, state whether the ring is commutative, whether it has unity, and whether it is a field. $$ \mathrm{Z} \text { " with the usual addition and multiplication. } $$
Problem 9
In Exercises 7 through 13, decide whether the indicated operations of addition and multiplication are defined (closed) on the set, and give a ring structure. If a ring is not formed, tell why this is the case. If a ring is formed, state whether the ring is commutative, whether it has unity, and whether it is a field. \(\mathbb{Z} \times \mathbb{Z}\) with addition and multiplication by components
Problem 9
Decide whether the indicated operations of addition and multiplication are defined (closed) on the set, and give a ring structure. If a ring is not formed, tell why this is the case. If a ring is formed, state whether the ring is commutative, whether it has unity, and whether it is a field. $$ \mathbb{Z} \times \mathbb{Z} \text { with addition and multiplication by components } $$
Problem 10
In Exercises 7 through 13, decide whether the indicated operations of addition and multiplication are defined (closed) on the set, and give a ring structure. If a ring is not formed, tell why this is the case. If a ring is formed, state whether the ring is commutative, whether it has unity, and whether it is a field. \(2 Z \times 2\) with addition and multiplication by components
Problem 10
Decide whether the indicated operations of addition and multiplication are defined (closed) on the set, and give a ring structure. If a ring is not formed, tell why this is the case. If a ring is formed, state whether the ring is commutative, whether it has unity, and whether it is a field. $$ 2 Z \times \mathbb{Z} \text { with addition and multiplication by components } $$
Problem 11
In Exercises 7 through 13, decide whether the indicated operations of addition and multiplication are defined (closed) on the set, and give a ring structure. If a ring is not formed, tell why this is the case. If a ring is formed, state whether the ring is commutative, whether it has unity, and whether it is a field. \((a+b \sqrt{2} \mid a, b \in \mathbb{Z}\\}\) with the usual addition and multiplication
Problem 11
Decide whether the indicated operations of addition and multiplication are defined (closed) on the set, and give a ring structure. If a ring is not formed, tell why this is the case. If a ring is formed, state whether the ring is commutative, whether it has unity, and whether it is a field. $$ \\{a+b \sqrt{2} \mid a, b \in \mathbb{Z}) \text { with the usual addition and multiplication } $$
Problem 12
Decide whether the indicated operations of addition and multiplication are defined (closed) on the set, and give a ring structure. If a ring is not formed, tell why this is the case. If a ring is formed, state whether the ring is commutative, whether it has unity, and whether it is a field. $$ (a+b \sqrt{2} \mid a, b \in \mathbb{Q}) \text { with the usual addition and multiplication } $$
Problem 13
Decide whether the indicated operations of addition and multiplication are defined (closed) on the set, and give a ring structure. If a ring is not formed, tell why this is the case. If a ring is formed, state whether the ring is commutative, whether it has unity, and whether it is a field. The set of all pure imaginary complex numbers \(r i\) for \(r \in \mathbb{R}\) with the usual addition and multiplication
Problem 14
In Exercises 14 through 19 , describe all units in the given ring \(\mathbf{z}\)
