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In abstract algebra, an element a of a ring R is called a left zero divisor if there exists a nonzero x in R such that ax = 0, [1] or equivalently if the map from R to R that sends x to ax is not injective. [a] Similarly, an element a of a ring is called a right zero divisor if there exists a nonzero y in R such that ya = 0.
If one interprets the definition of divisor literally, every a is a divisor of 0, since one can take x = 0. Because of this, it is traditional to abuse terminology by making an exception for zero divisors: one calls an element a in a commutative ring a zero divisor if there exists a nonzero x such that ax = 0. [2]
Let R be an effective commutative ring.. There is an algorithm for testing if an element a is a zero divisor: this amounts to solving the linear equation ax = 0.; There is an algorithm for testing if an element a is a unit, and if it is, computing its inverse: this amounts to solving the linear equation ax = 1.
For example, using single-precision IEEE arithmetic, if x = −2 −149, then x/2 underflows to −0, and dividing 1 by this result produces 1/(x/2) = −∞. The exact result −2 150 is too large to represent as a single-precision number, so an infinity of the same sign is used instead to indicate overflow.
A positive divisor of that is different from is called a proper divisor or an aliquot part of (for example, the proper divisors of 6 are 1, 2, and 3). A number that does not evenly divide n {\displaystyle n} but leaves a remainder is sometimes called an aliquant part of n . {\displaystyle n.}
The zero ring is the unique ring in which the additive identity 0 and multiplicative identity 1 coincide. [1] [6] (Proof: If 1 = 0 in a ring R, then for all r in R, we have r = 1r = 0r = 0. The proof of the last equality is found here.) The zero ring is commutative. The element 0 in the zero ring is a unit, serving as its own multiplicative ...
Because the canonical divisor is intrinsically associated to a variety, a key role in the classification of varieties is played by the maps to projective space given by K X and its positive multiples. The Kodaira dimension of X is a key birational invariant, measuring the growth of the vector spaces H 0 (X, mK X) (meaning H 0 (X, O(mK X))) as m ...
In abstract algebra, given a magma with binary operation ∗ (which could nominally be termed multiplication), left division of b by a (written a \ b) is typically defined as the solution x to the equation a ∗ x = b, if this exists and is unique. Similarly, right division of b by a (written b / a) is the solution y to the equation y ∗ a = b ...