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. Thank you.
To 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, we need to show that each statement implies the other.
1. Cancellation Law implies that the product of non-zero factors is not zero:
Assume that the cancellation law of multiplication holds in a commutative ring. This means that if we have any elements a, b, and c in the ring such that a ≠ 0 and ab = ac, then we can cancel a from both sides of the equation to obtain b = c.
Now let's consider the product of non-zero factors: ab, where a, b ≠ 0. If ab = 0, then we can apply the cancellation law by multiplying both sides of the equation by a^(-1), the multiplicative inverse of a, to get a^(-1)(ab) = a^(-1)0, which simplifies to (a^(-1)a)b = 0. Since a^(-1)a = 1 (the identity element of the ring), we have 1b = 0. But this contradicts the assumption that b ≠ 0, as the product of any element with the identity element is that element itself, not zero. Thus, the product of non-zero factors is not zero.
2. The product of non-zero factors is not zero implies the Cancellation Law:
Let's assume that the product of non-zero factors is not zero in a commutative ring. Now we need to show that the cancellation law holds.
Suppose we have elements a, b, and c in the ring such that a ≠ 0 and ab = ac. If we can prove that b = c, then the cancellation law holds.
We start by multiplying both sides of the equation ab = ac by a^(-1), the multiplicative inverse of a. This gives us (a^(-1))(ab) = (a^(-1))(ac), which simplifies to (a^(-1)a)b = (a^(-1)a)c. Since a^(-1)a = 1, we have 1b = 1c, which implies that b = c. Therefore, the cancellation law holds.
Thus, we have shown 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.