# 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
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.

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

### Relevant Questions

Suppose that R is a commutative ring without zero-divisors. Show that all the nonzero elements of R have the same additive order.
Let R and S be commutative rings. Prove that (a, b) is a zero-divisor in $$\displaystyle{R}\oplus{S}$$ if and only if a or b is a zero-divisor or exactly one of a or b is 0.
Suppose that R and S are commutative rings with unites, Let PSJphiZSK be a ring homomorphism from R onto S and let A be an ideal of S.
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Suppose that R is a ring and that $$\displaystyle{a}^{{2}}={a}$$ for all $$\displaystyle{a}\in{R}{Z}$$. Show that R is commutative.
Let a belong to a ring R. Let $$\displaystyle{S}={\left\lbrace{x}\in{R}{\mid}{a}{x}={0}\right\rbrace}$$ . Show that S is a subring of R.