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

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.