91Ó°ÊÓ

The concept of closure under addition involves checking whether the sum of any two elements from a given set also belongs to the same set. In our exercise, the set is defined as all numbers of the form \( a + b\sqrt{2} \), where both \( a \) and \( b \) are rational numbers. To determine closure under addition, take two elements from this set, say \( x = a_1 + b_1\sqrt{2} \) and \( y = a_2 + b_2\sqrt{2} \).
We calculate their sum:

• \( x + y = (a_1 + a_2) + (b_1 + b_2)\sqrt{2} \)

Since the sum of any two rational numbers \( a_1 + a_2 \) and \( b_1 + b_2 \) is also rational, the result \( (a_1 + a_2) + (b_1 + b_2)\sqrt{2} \) remains within the set. Therefore, the set is closed under addition. This property is crucial because it ensures that when you add elements of the set, you don't "step outside" of it.

The closure under multiplication requires that when two elements from a set are multiplied, the result must also be an element of the same set. For the set \( S = \{ a + b\sqrt{2} \,|\, a, b \in \mathbb{Q} \} \), consider two elements \( x = a_1 + b_1\sqrt{2} \) and \( y = a_2 + b_2\sqrt{2} \). Their product is calculated as:

• \( x \cdot y = (a_1a_2 + 2b_1b_2) + (a_1b_2 + a_2b_1)\sqrt{2} \)

It's important to check if this product fits the same form \( a + b\sqrt{2} \). Notice that \( a_1a_2 + 2b_1b_2 \) and \( a_1b_2 + a_2b_1 \) are both rational, as they are combinations of products and sums of rational numbers. Because this product \( (a_1a_2 + 2b_1b_2) + (a_1b_2 + a_2b_1)\sqrt{2} \) still matches the form \( a + b\sqrt{2} \), the set maintains closure under multiplication.
This property is vital for sets being candidates for rings, ensuring no multiplication steps out of the defined set boundaries.

In ring theory, a commutative ring is a ring in which the multiplication operation is commutative; that is, the order in which two elements are multiplied does not affect the result. For the set \( S \) defined by \( a + b\sqrt{2} \), let's check both addition and multiplication for commutativity. Consider two elements, \( x = a_1 + b_1\sqrt{2} \) and \( y = a_2 + b_2\sqrt{2} \).

• Adding \( x + y \) is equivalent to \( y + x \) due to the commutative property of addition over rational numbers.
• Multiplication \( x \cdot y = y \cdot x \) follows from the commutative property of multiplying rational numbers.

This commutative property holds for both operations for the set, confirming that our ring is indeed a commutative ring.
Commutative rings are significant as they simplify many algebraic structures and make certain theoretical and applied problems more manageable.

The ring axioms are a set of rules that a structure must satisfy to be considered a ring. Our set \( S = \{ a + b\sqrt{2} \,|\, a, b \in \mathbb{Q} \} \) needs to be checked against these axioms, which include:

• Closure: As shown, \( S \) is closed under both addition and multiplication.
• Additive Identity: The element \( 0 + 0\sqrt{2} \) serves as the identity.
• Additive Inverse: For any \( a + b\sqrt{2} \) in \( S \), the inverse is \( -a - b\sqrt{2} \).
• Associativity: Usual addition and multiplication are associative.
• Distributivity: Regular distributive laws apply.

Finally, our set must be validated for a multiplicative identity (unity), provided by \( 1 + 0\sqrt{2} \), but it fails to be a field, as not every element's inverse stays within the set.
These axioms collectively provide a robust framework ensuring the set's algebraic structure is a ring, streamlining operations and applications within the framework of ring theory.