Is Every Element in a Finite Commutative Ring a Unit or Zero-Divisor?

Thread starter xixi
Start date Jun 30, 2010

In summary, a non-unit in mathematics is an element in a ring that cannot be multiplied by another element to result in the identity element. This makes it a zero-divisor. A zero-divisor, on the other hand, is an element that, when multiplied by another element, results in the additive identity element. A non-unit is always a zero-divisor, but a zero-divisor cannot be a unit because it does not have a multiplicative inverse. This is because a unit must have a multiplicative inverse, while a zero-divisor does not. A non-unit is considered a zero-divisor because its product with another element is equal to zero, meeting the definition of a zero-divisor in mathematics.

Jun 30, 2010

xixi

Let R be a finite commutative ring . Then show that each element of R is a unit or a zero-divisor .

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Jun 30, 2010

Hurkyl

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Related to Is Every Element in a Finite Commutative Ring a Unit or Zero-Divisor?

1. What is a non-unit in mathematics?

A non-unit in mathematics refers to any element in a ring that is not invertible, meaning it does not have a multiplicative inverse. This means that the element cannot be multiplied by another element to result in the identity element.

2. What is a zero-divisor in mathematics?

A zero-divisor in mathematics refers to any element in a ring that, when multiplied by another element, results in the additive identity element (zero). This means that the product of the two elements is equal to zero, but neither element is equal to zero individually.

3. How is a non-unit related to a zero-divisor?

A non-unit in a ring will always be a zero-divisor, as it cannot be multiplied by any other element to result in the identity element. This means that a non-unit will always have a product with another element that is equal to zero.

4. Can a zero-divisor be a unit in a ring?

No, a zero-divisor cannot be a unit in a ring. This is because a unit must have a multiplicative inverse, meaning it can be multiplied by another element to result in the identity element. However, a zero-divisor does not have a multiplicative inverse and thus cannot be a unit.

5. Why is a non-unit considered a zero-divisor?

A non-unit is considered a zero-divisor because it cannot be multiplied by any element to result in the identity element, and thus its product with another element is equal to zero. This meets the definition of a zero-divisor in mathematics.