Interface DivisionRing<E>

All Superinterfaces:
AbelianGroup<E>, AbelianMonoid<E>, Group<E>, Magma<E>, Monoid<E>, Ring<E>, Semiring<E>, Set<E>
public interface DivisionRing<E> extends Ring<E>
A division ring (also known as a skew field) is a ring in which every non-zero element has a multiplicative inverse.

Division rings satisfy all field axioms except for commutative multiplication.

Mathematical Definition

A division ring (D, +, ×) is a ring with unity (1 ≠ 0) such that:

  • For every a ∈ D \ {0}, there exists a⁻¹ ∈ D such that a × a⁻¹ = a⁻¹ × a = 1

Examples

  • ℍ - Quaternions (non-commutative)
  • Field - Commutative division ring
Since:
1.0
Author:
Silvere Martin-Michiellot, Gemini AI (Google DeepMind)

  • Method Details

    • inverse

      E inverse(E element)
      Returns the multiplicative inverse of a non-zero element.

      For element a ≠ 0, returns a⁻¹ such that: a × a⁻¹ = a⁻¹ × a = 1

      Specified by:
      inverse in interface Group<E>
      Parameters:
      element - the element to invert (must be non-zero)
      Returns:
      the multiplicative inverse
      Throws:
      ArithmeticException - if element is zero

    • divide

      default E divide(E dividend, E divisor)
      Returns the quotient of two elements (division).

      Defined as: a ÷ b = a × b⁻¹

      Note: In non-commutative rings, this is typically defined as right division. For left division (b⁻¹ × a), explicit multiplication by inverse is recommended.

      Parameters:
      dividend - the dividend (numerator)
      divisor - the divisor (denominator, must be non-zero)
      Returns:
      dividend ÷ divisor
      Throws:
      ArithmeticException - if divisor is zero