Task:

A. Use the definition for a ring to prove that Z7 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.

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.