Rings are an important algebraic structure, and modular arithmetic has that structure.

Recall that for the mod m relation, the congruence class of an integer x is denoted [x]m. For example, the elements of [–5]7 are of the form –5 plus integer multiples of 7, which would equate to {. . . –19, –12, –5, 2, 9, 16, . . .} or, more formally, {y: y = -5 + 7q for some integer q}.

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.