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Two divisors are said to be linearly equivalent if their difference is principal, so the divisor class group is the group of divisors modulo linear equivalence. For a variety X of dimension n over a field, the divisor class group is a Chow group ; namely, Cl( X ) is the Chow group CH n −1 ( X ) of ( n −1)-dimensional cycles.
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.
When R is commutative, the notions of left divisor, right divisor, and two-sided divisor coincide, so one says simply that a is a divisor of b, or that b is a multiple of a, and one writes . Elements a and b of an integral domain are associates if both a ∣ b {\displaystyle a\mid b} and b ∣ a {\displaystyle b\mid a} .
The hypothesis that the geometric genus is positive essentially means (by the Lefschetz theorem on (1,1)-classes) that the cohomology group () contains transcendental information, and in effect Mumford's theorem implies that, despite having a purely algebraic definition, it shares transcendental information with (). Mumford's theorem has ...
A linear system of divisors algebraicizes the classic geometric notion of a family of curves, as in the Apollonian circles.. In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family.
A has a divisor theory in which every divisor is principal. A is a Krull domain in which every divisorial ideal is principal (in fact, this is the definition of UFD in Bourbaki.) A is a Krull domain and every prime ideal of height 1 is principal. [7] In practice, (2) and (3) are the most useful conditions to check.
If p is a prime number which is not a divisor of b, then ab p−1 mod p = a mod p, due to Fermat's little theorem. Inverse: [(−a mod n) + (a mod n)] mod n = 0. b −1 mod n denotes the modular multiplicative inverse, which is defined if and only if b and n are relatively prime, which is the case when the left hand side is defined: [(b −1 ...
An integral domain is a UFD if and only if it is a GCD domain (i.e., a domain where every two elements have a greatest common divisor) satisfying the ascending chain condition on principal ideals. An integral domain is a Bézout domain if and only if any two elements in it have a gcd that is a linear combination of the two.