Ring Verification: Integer And Rational Number Sets

by Tim Redaksi 52 views
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Let's dive into determining whether given sets with their defined operations form a ring. We'll explore two specific cases: a set of 2x2 matrices with integer entries and the set of all rational numbers. Buckle up, math enthusiasts!

a. Verifying M as a Ring

We're given the set M, which consists of 2x2 matrices of the form pmatrix a & 0 \ b & 0 pmatrix, where a and b are integers. We need to check if (M, +, ×) forms a ring under standard matrix addition (+) and matrix multiplication (×).

To prove that M is a ring, we must verify the following ring axioms:

  1. (M, +) is an abelian group:
    • Closure under addition: For any two matrices in M, their sum must also be in M.
    • Associativity of addition: Matrix addition is associative.
    • Existence of additive identity: There must be a zero matrix in M.
    • Existence of additive inverse: For every matrix in M, there must be an inverse matrix in M such that their sum is the zero matrix.
    • Commutativity of addition: Matrix addition must be commutative.
  2. (M, ×) is a semigroup:
    • Closure under multiplication: For any two matrices in M, their product must also be in M.
    • Associativity of multiplication: Matrix multiplication is associative.
  3. Distributive laws:
    • Left distributive law: A × (B + C) = (A × B) + (A × C) for all A, B, C in M.
    • Right distributive law: (A + B) × C = (A × C) + (B × C) for all A, B, C in M.

Let A = pmatrix a & 0 \ b & 0 pmatrix and B = pmatrix c & 0 \ d & 0 pmatrix be two arbitrary elements in M, where a, b, c, and d are integers.

Verifying (M, +) as an Abelian Group

  • Closure under addition:

    A + B = pmatrix a+c & 0 \ b+d & 0 pmatrix. Since a+c and b+d are integers, A + B is also in M. Therefore, M is closed under addition.

  • Associativity of addition:

    Matrix addition is associative in general, so it holds for M.

  • Existence of additive identity:

    The zero matrix 0 = pmatrix 0 & 0 \ 0 & 0 pmatrix is in M (since 0 is an integer). For any A in M, A + 0 = A. Thus, the additive identity exists in M.

  • Existence of additive inverse:

    For A = pmatrix a & 0 \ b & 0 pmatrix, the additive inverse is -A = pmatrix -a & 0 \ -b & 0 pmatrix. Since -a and -b are integers, -A is also in M. A + (-A) = 0. Thus, the additive inverse exists for every element in M.

  • Commutativity of addition:

    A + B = pmatrix a+c & 0 \ b+d & 0 pmatrix = pmatrix c+a & 0 \ d+b & 0 pmatrix = B + A. Thus, matrix addition is commutative in M.

Since all the conditions are met, (M, +) is an abelian group.

Verifying (M, ×) as a Semigroup

  • Closure under multiplication:

    A × B = pmatrix a & 0 \ b & 0 pmatrix × pmatrix c & 0 \ d & 0 pmatrix = pmatrix a*c & 0 \ b*c & 0 pmatrix. Since ac and bc are integers, A × B is also in M. Therefore, M is closed under multiplication.

  • Associativity of multiplication:

    Matrix multiplication is associative in general, so it holds for M.

Since both conditions are met, (M, ×) is a semigroup.

Verifying Distributive Laws

Let C = pmatrix e & 0 \ f & 0 pmatrix be another element in M, where e and f are integers.

  • Left distributive law:

    A × (B + C) = pmatrix a & 0 \ b & 0 pmatrix × pmatrix c+e & 0 \ d+f & 0 pmatrix = pmatrix a*(c+e) & 0 \ b*(c+e) & 0 pmatrix = pmatrix a*c + a*e & 0 \ b*c + b*e & 0 pmatrix

    (A × B) + (A × C) = pmatrix a*c & 0 \ b*c & 0 pmatrix + pmatrix a*e & 0 \ b*e & 0 pmatrix = pmatrix a*c + a*e & 0 \ b*c + b*e & 0 pmatrix

    Thus, A × (B + C) = (A × B) + (A × C).

  • Right distributive law:

    (A + B) × C = pmatrix a+c & 0 \ b+d & 0 pmatrix × pmatrix e & 0 \ f & 0 pmatrix = pmatrix (a+c)*e & 0 \ (b+d)*e & 0 pmatrix = pmatrix a*e + c*e & 0 \ b*e + d*e & 0 pmatrix

    (A × C) + (B × C) = pmatrix a*e & 0 \ b*e & 0 pmatrix + pmatrix c*e & 0 \ d*e & 0 pmatrix = pmatrix a*e + c*e & 0 \ b*e + d*e & 0 pmatrix

    Thus, (A + B) × C = (A × C) + (B × C).

Since the distributive laws hold, we can conclude that (M, +, ×) is a ring.

b. Analyzing R (Rational Numbers)

The question statement is incomplete and does not provide any defined operations for the set R (rational numbers). To determine if R is a ring, we need to know what operations are defined on it. However, assuming the standard addition (+) and multiplication (×) operations for rational numbers, we can proceed as follows:

Assuming Standard Operations: (R, +, ×)

If we consider R with standard addition and multiplication, we can verify the ring axioms:

  1. (R, +) is an abelian group:
    • Closure under addition: The sum of two rational numbers is always a rational number.
    • Associativity of addition: Addition of rational numbers is associative.
    • Existence of additive identity: 0 is a rational number, and it serves as the additive identity.
    • Existence of additive inverse: For any rational number a, -a is also a rational number and is its additive inverse.
    • Commutativity of addition: Addition of rational numbers is commutative.
  2. (R, ×) is a semigroup:
    • Closure under multiplication: The product of two rational numbers is always a rational number.
    • Associativity of multiplication: Multiplication of rational numbers is associative.
  3. Distributive laws:
    • Multiplication distributes over addition for rational numbers.

Since all ring axioms are satisfied under standard addition and multiplication, (R, +, ×) is a ring.

Furthermore, since multiplication of rational numbers is also commutative, (R, +, ×) is a commutative ring. Also, since the multiplicative identity 1 is a rational number, (R, +, ×) is a ring with unity.

Conclusion

  • For the set M of 2x2 matrices with integer entries as defined, (M, +, ×) is a ring.
  • For the set R of rational numbers, assuming standard addition and multiplication, (R, +, ×) is a ring, specifically a commutative ring with unity.

Math can be pretty cool, right? Keep exploring!