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  2. Additive inverse - Wikipedia

    en.wikipedia.org/wiki/Additive_inverse

    In a vector space, the additive inverse −v (often called the opposite vector of v) has the same magnitude as v and but the opposite direction. [11] In modular arithmetic, the modular additive inverse of x is the number a such that a + x ≡ 0 (mod n) and always exists. For example, the inverse of 3 modulo 11 is 8, as 3 + 8 ≡ 0 (mod 11). [12]

  3. Ring (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Ring_(mathematics)

    The axioms of modules imply that (−1)x = −x, where the first minus denotes the additive inverse in the ring and the second minus the additive inverse in the module. Using this and denoting repeated addition by a multiplication by a positive integer allows identifying abelian groups with modules over the ring of integers.

  4. Unit (ring theory) - Wikipedia

    en.wikipedia.org/wiki/Unit_(ring_theory)

    The multiplicative identity 1 and its additive inverse −1 are always units. More generally, any root of unity in a ring R is a unit: if r n = 1, then r n−1 is a multiplicative inverse of r. In a nonzero ring, the element 0 is not a unit, so R × is not closed under addition.

  5. Inverse element - Wikipedia

    en.wikipedia.org/wiki/Inverse_element

    For example, addition is a total associative operation on nonnegative integers, which has 0 as additive identity, and 0 is the only element that has an additive inverse. This lack of inverses is the main motivation for extending the natural numbers into the integers.

  6. Algebraic structure - Wikipedia

    en.wikipedia.org/wiki/Algebraic_structure

    For example, in the case of numbers, the additive inverse is provided by the unary minus operation . Also, in universal algebra , a variety is a class of algebraic structures that share the same operations, and the same axioms, with the condition that all axioms are identities.

  7. Idempotent (ring theory) - Wikipedia

    en.wikipedia.org/wiki/Idempotent_(ring_theory)

    Some authors use the term "idempotent ring" for this type of ring. In such a ring, multiplication is commutative and every element is its own additive inverse. A ring is semisimple if and only if every right (or every left) ideal is generated by an idempotent.

  8. Additive combinatorics - Wikipedia

    en.wikipedia.org/wiki/Additive_combinatorics

    Additive combinatorics is an area of combinatorics in mathematics. One major area of study in additive combinatorics are inverse problems: ... Examples of this type ...

  9. Inequality (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Inequality_(mathematics)

    The rules for the additive inverse, and the multiplicative inverse for positive numbers, are both examples of applying a monotonically decreasing function. If the inequality is strict ( a < b , a > b ) and the function is strictly monotonic, then the inequality remains strict.