Question

In: Advanced Math

Suppose a and b belong to a commutative ring and ab is a zero-divisor. Show that...

Suppose a and b belong to a commutative ring and ab is a zero-divisor. Show that either a or b is a zero-divisor. provide explanation please.

Expert Solution

Recall the following defintions:

Let be a ring.

(i) is called commutative if .

(ii) An element is called a zero divisor if there is a non-zero element such that either or .

Back to the problem:

Suppose is a commutative ring and such that is a zero divisor.

Then there exists a non-zero element such that either or . Here both the conditions are same because of the commutativity.

If is non-zero then from the definition of a zero divisor it follows that is a zero divisor.

If then from the definition of a zero divisor it follows that is a zero divisor because here is non-zero. (Proved)

Colby Messinger answered 1 year ago

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Suppose a and b belong to a commutative ring and ab is a zero-divisor. Show that...

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