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Show that a finite commutative ring with no zero-divisors
Chapter 13, Problem 45E(choose chapter or problem)
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