A. Use the definition for a ring to prove that Z

_{7}is a ring under the operations + and × defined as follows: [a]

_{7}+ [b]_{7}= [a + b]_{7}and [a]_{7}× [b]_{7}= [a × b]_{7}Note: On the right-hand-side of these equations, + and × are the usual operations on the integers, so the modular versions of addition and multiplication inherit many properties from integer addition and multiplication.

1. State each step of your proof.

2. Provide written justification for each step of your proof.

By demonstration of the above properties, Z

_{7}meets all of the criteria of a ring.B. Use the definition for an integral domain to prove that Z7 is an integral domain.

1. State each step of your proof.

2. Provide written justification for each step of your proof.