Proving Z[√2] is a Ring: A Comprehensive Guide for SEO

In the vast world of abstract algebra, the concept of a ring is a fundamental building block. A ring is a set equipped with two binary operations, addition and multiplication, that satisfy certain properties. To understand how we can show that Z[√2] is a ring, we first need to comprehend the definition and properties of a ring. This article aims to guide you through the process of proving that Z[√2] is indeed a ring, ensuring that your content is optimized for search engines like Google while providing a rich, informative article for readers.

Defining a Ring and Its Properties

Before diving into the proof, let's start by defining a ring more formally. A ring R is a set equipped with two binary operations, addition ( ) and multiplication (·), such that:

Abelian Group under Addition: The set R forms an abelian group under addition. This means: Closure under Addition: For all a, b in R, a b is also in R. Associativity of Addition: For all a, b, c in R, (a b) c a (b c). Commutativity of Addition: For all a, b in R, a b b a. Additive Identity: There exists an element 0 in R such that for all a in R, a 0 a. Additive Inverses: For each a in R, there exists an element -a in R such that a (-a) 0. Multiplicative Monoid: The set R forms a monoid under multiplication. This means: Closure under Multiplication: For all a, b in R, a · b is also in R. Associativity of Multiplication: For all a, b, c in R, (a · b) · c a · (b · c). Multiplicative Identity: There exists an element 1 in R such that for all a in R, a · 1 a. Distributivity of Multiplication over Addition: For all a, b, c in R, a · (b c) a · b a · c and (a b) · c a · c b · c.

These properties clearly outline the structure of a ring and ensure that our set behaves in a consistent and predictable manner under the operations defined.

Understanding Z[√2]

Now, let's focus on the set in question: Z[√2]. This set consists of all elements of the form ab√2 ab where a, b are integers (Z). To show that Z[√2] is a ring, we need to verify that it satisfies the properties of a ring as defined above.

Proving Z[√2] is a Subring of Real Numbers

Given that the set of real numbers (R) is already a known ring, we can use this fact to simplify our task. A subring of a ring R is a subset S of R that is itself a ring under the same operations as R. Therefore, to show that Z[√2] is a subring of the real numbers, we need to demonstrate that Z[√2] is closed under both addition and multiplication, and that it contains the multiplicative identity 1.

1. Closure Under Addition

Let's take any two elements from Z[√2], say (a?b?√2 a?b?) and (a?b?√2 a?b?). We need to show that their sum is also in Z[√2].

(a?b?√2 a?b?) (a?b?√2 a?b?) (a?b? a?b?)√2 (a?b? a?b?)

Since the sum of two integers (a?b? a?b?) is still an integer, this operation confirms that Z[√2] is closed under addition.

2. Closure Under Multiplication

Now, let's consider the multiplication of two elements in Z[√2].

(a?b?√2 a?b?) · (a?b?√2 a?b?) (a?b?a?b?) · 2 (a?b?a?b?) (a?b?a?b?√2 a?b?a?b?)

Simplifying, we get:

(a?b?a?b?) · 2 (a?b?a?b?) (a?b?a?b?√2 a?b?a?b?) (a?a?)(b?b?)(2√2 1) (a?a?)(b?b?√2 b?b?)

Again, the result is in the form ab√2 ab, confirming that Z[√2] is closed under multiplication.

3. Contains the Additive Identity

The element 0 is in Z[√2] because:

0 0√2 0

which fits the form ab√2 ab with a 0 and b 0.

4. Contains the Multiplicative Identity

The element 1 is in Z[√2] because:

1 1√2 1

which also fits the form ab√2 ab with a 1 and b 0.

Conclusion

We have now shown that Z[√2] satisfies all the necessary conditions to be a subring of the real numbers. Therefore, we can conclude that Z[√2] is indeed a ring, satisfying the abelian group under addition, the monoid under multiplication, and the distributive property.

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