Ad
related to: additive inverse formula examples with solutions video youtube free
Search results
Results From The WOW.Com Content Network
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]
Thus, for example, the only fields for which the rank of the free part is zero are Q and the imaginary quadratic fields. A more precise statement giving the structure of O × ⊗ Z Q as a Galois module for the Galois group of K/Q is also possible. [14] The free part of the unit group can be studied using the infinite places of K. Consider the ...
Exponentiation to negative integers can be further extended to invertible elements of a ring by defining x −1 as the multiplicative inverse of x; in this context, these elements are considered units. [1]: p.49 In a polynomial domain F [x] over any field F, the polynomial x has no inverse. If it did have an inverse q(x), then there would be [5]
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.
One may consider the ring of integers modulo n, where n is square-free. By the Chinese remainder theorem, this ring factors into the product of rings of integers modulo p, where p is prime. Now each of these factors is a field, so it is clear that the factors' only idempotents will be 0 and 1. That is, each factor has two idempotents.
In algebra, a unit or invertible element [a] of a ring is an invertible element for the multiplication of the ring. That is, an element u of a ring R is a unit if there exists v in R such that = =, where 1 is the multiplicative identity; the element v is unique for this property and is called the multiplicative inverse of u.
Any involution is a bijection.. The identity map is a trivial example of an involution. Examples of nontrivial involutions include negation (x ↦ −x), reciprocation (x ↦ 1/x), and complex conjugation (z ↦ z) in arithmetic; reflection, half-turn rotation, and circle inversion in geometry; complementation in set theory; and reciprocal ciphers such as the ROT13 transformation and the ...
In algebra, an additive map, -linear map or additive function is a function that preserves the addition operation: [1] (+) = + for every pair of elements and in the domain of . For example, any linear map is additive.