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Show that a finite commutative ring with no zero-divisors

Chapter 13, Problem 45E

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QUESTION:

Show that a finite commutative ring with no zero-divisors and at least two elements has a unity.

Questions & Answers

QUESTION:

Show that a finite commutative ring with no zero-divisors and at least two elements has a unity.

ANSWER:

Step 1 of 2

Let R be a finite commutative ring with no zero divisors and has at least two elements.

To prove that R has a unity.

Let be the set of non zero elements in R.

First, let us prove that are all distinct.

Suppose,

Since R is a ring with no zero divisors and , we have

Then, the set S can be written as

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