**Task:**

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.

B. Use the definition for an integral domain to prove that *Z*_{7} is an integral domain.

1. State *each* step of your proof.

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

Subject | Mathematics |

Due By (Pacific Time) | 12/10/2014 12:00 am |

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