Prove that the cancellation law of multiplication is equivalent in a commutative ring to the assertion that the product of non-zero factors is not zero
To prove that the cancellation law of multiplication is equivalent to the assertion that the product of non-zero factors is not zero in a commutative ring, we need to demonstrate both implications.
First, let's assume that the cancellation law of multiplication holds in a commutative ring. This law states that for any elements a, b, and c in the ring, if a * b = a * c and a is non-zero, then b = c. We want to show that the product of non-zero factors is not zero.
Let's assume that a and b are non-zero elements in the commutative ring, and their product, a * b, is equal to zero. The cancellation law of multiplication implies that since a * b = a * 0 (zero element), and a is non-zero, we can cancel a from both sides, yielding b = 0.
However, this contradicts our assumption that b is non-zero. Hence, we have proven that if the cancellation law of multiplication holds in a commutative ring, the product of non-zero factors is not zero.
Now, let's demonstrate the other implication. We assume that the product of non-zero factors is not zero in a commutative ring and want to prove the cancellation law of multiplication.
Let's consider three elements a, b, and c in the ring, where a is non-zero, and a * b = a * c. We need to show that b = c.
Suppose that a * b = a * c. Since a is non-zero, the product of non-zero factors is not zero. Therefore, a * b = a * c implies that b = c.
Hence, we have proven that if the product of non-zero factors is not zero in a commutative ring, then the cancellation law of multiplication holds.
In summary, we have shown both implications:
1. If the cancellation law of multiplication holds, then the product of non-zero factors is not zero.
2. If the product of non-zero factors is not zero, then the cancellation law of multiplication holds.
Therefore, the cancellation law of multiplication is equivalent to the assertion that the product of non-zero factors is not zero in a commutative ring.