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Integer multiplication respects the congruence classes, that is, a ≡ a' and b ≡ b' (mod n) implies ab ≡ a'b' (mod n). This implies that the multiplication is associative, commutative, and that the class of 1 is the unique multiplicative identity. Finally, given a, the multiplicative inverse of a modulo n is an integer x satisfying ax ≡ ...
A commutative semigroup is a set endowed with a total, associative and commutative operation. If the operation additionally has an identity element, we have a commutative monoid; An abelian group, or commutative group is a group whose group operation is commutative. [16] A commutative ring is a ring whose multiplication is commutative.
Likewise, the trivial operation x ∘ y = y (that is, the result is the second argument, no matter what the first argument is) is associative but not commutative. Addition and multiplication of complex numbers and quaternions are associative. Addition of octonions is also associative, but multiplication of octonions is non-associative.
The manipulations of the Rubik's Cube form the Rubik's Cube group.. In mathematics, a group is a set with an operation that associates every pair of elements of the set to an element of the set (as does every binary operation) and satisfies the following constraints: the operation is associative, it has an identity element, and every element of the set has an inverse element.
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product.Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear".
In general, matrices, even invertible matrices, do not form an abelian group under multiplication because matrix multiplication is generally not commutative. However, some groups of matrices are abelian groups under matrix multiplication – one example is the group of 2 × 2 {\displaystyle 2\times 2} rotation matrices .
However, matrix multiplication is not commutative, which shows that this group is non-abelian. Another fact worth noticing is that the integers under multiplication do not form a group—even if zero is excluded. This is easily seen by the nonexistence of an inverse for all elements other than 1 and −1.
In mathematics, an associative algebra A over a commutative ring (often a field) K is a ring A together with a ring homomorphism from K into the center of A.This is thus an algebraic structure with an addition, a multiplication, and a scalar multiplication (the multiplication by the image of the ring homomorphism of an element of K).
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