Ring Exercise 2

We wish to determine whether the function \( f(x) = |x| \) defines a ring homomorphism between the rings \( (\mathbb{Z}, +, \cdot) \) and \( (\mathbb{Z}', +, \cdot) \), where \( \mathbb{Z} = \mathbb{Z}' \).

We begin by checking whether \( f \) preserves multiplication, as required for a ring homomorphism:

$$ f(a \cdot b) = f(a) \cdot f(b) $$

$$ |a \cdot b| = |a| \cdot |b| $$

This equality always holds, since the absolute value of a product equals the product of the absolute values.

Note: On the left-hand side, the absolute value of the product is always non-negative. On the right-hand side, the product of two non-negative numbers (i.e., the absolute values) is also non-negative. Therefore, the two expressions agree.

Next, we examine whether \( f \) preserves addition:

$$ f(a + b) = f(a) + f(b) $$

$$ |a + b| = |a| + |b| $$

This identity does not hold in general—specifically, it fails whenever \( a \) and \( b \) have opposite signs.

For instance, consider \( a = 2 \) and \( b = -3 \):

$$ |2 + (-3)| \ne |2| + |-3| $$

$$ |-1| \ne 2 + 3 $$

$$ 1 \ne 5 $$

Since the function fails to preserve addition, it does not satisfy the defining properties of a ring homomorphism.

We conclude that the function \( f(x) = |x| \) is not a ring homomorphism from \( (\mathbb{Z}, +, \cdot) \) to \( (\mathbb{Z}', +, \cdot) \).

And so forth.