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.

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.

Question
Commutative Algebra
asked 2020-11-08
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.

Answers (1)

2020-11-09
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 \(\displaystyle{x}\ne{0}\)
Now, by associativity and commutatibity property of a ring, we have,
a(bx)=0
And
b(ax)=0
If, \(\displaystyle{0}\ne{b}{x}\in{R}\), 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 \(\displaystyle{x}\ne{0}\), 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
0

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