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

lwfrgin 2020-11-08 Answered
Suppose that a and b belong to a commutative ring and ab is a zero-divisor. Show that either a or b is a zero-divisor.
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Delorenzoz
Answered 2020-11-09 Author has 91 answers
Let R be a commutative ring. it is known that for ab to be a zero divisor of a ring, it means that, (ab)x=0 for some x in the ring R such that x0
Now, by associativity and commutatibity property of a ring, we have,
a(bx)=0
And
b(ax)=0
If, 0bxR, then we can say by definition of zezro divisor that a is zero divisor. Also, on the other hand, if we have bx=0, then as the value of x0, by our assumption, get again by using the definition of zero divisor that, b is a zero divisor
Hence, it can be a concluded that either a or b is a zero-divisor
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