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For every x except 0, y represents its multiplicative inverse. The graph forms a rectangular hyperbola. In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x −1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the ...
The inverse or multiplicative inverse (for avoiding confusion with additive inverses) of a unit x is denoted , or, when the multiplication is commutative, . The additive identity 0 is never a unit, except when the ring is the zero ring , which has 0 as its unique element.
The element 0 in the zero ring is a unit, serving as its own multiplicative inverse. The unit group of the zero ring is the trivial group {0}. The element 0 in the zero ring is not a zero divisor. The only ideal in the zero ring is the zero ideal {0}, which is
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.
A rng of square zero is a rng R such that xy = 0 for all x and y in R. [4] Any abelian group can be made a rng of square zero by defining the multiplication so that xy = 0 for all x and y; [5] thus every abelian group is the additive group of some rng. The only rng of square zero with a multiplicative identity is the zero ring {0}. [5]
Books on commutative algebra or algebraic geometry often adopt the convention that ring means commutative ring, to simplify terminology. In a ring, multiplicative inverses are not required to exist. A nonzero commutative ring in which every nonzero element has a multiplicative inverse is called a field.
For associative algebras, the definition can be simplified as follows: a non-zero associative algebra over a field is a division algebra if and only if it has a multiplicative identity element 1 and every non-zero element a has a multiplicative inverse (i.e. an element x with ax = xa = 1).
But in the ring Z/6Z, 2 is a zero divisor. This equation has two distinct solutions, x = 1 and x = 4, so the expression is undefined. In field theory, the expression is only shorthand for the formal expression ab −1, where b −1 is the multiplicative inverse of b.