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A Riemann surface is a 1-dimensional complex manifold, and so its codimension-1 submanifolds have dimension 0. The group of divisors on a compact Riemann surface X is the free abelian group on the points of X. Equivalently, a divisor on a compact Riemann surface X is a finite linear combination of points of X with integer coefficients.
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.
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.
Since the field axioms only guarantee the existence of such inverses for nonzero elements, this expression has no meaning when b is zero. Modern texts, that define fields as a special type of ring, include the axiom 0 ≠ 1 for fields (or its equivalent) so that the zero ring is excluded from being a field. In the zero ring, division by zero is ...
In algebra, a domain is a nonzero ring in which ab = 0 implies a = 0 or b = 0. [1] ( Sometimes such a ring is said to "have the zero-product property".) Equivalently, a domain is a ring in which 0 is the only left zero divisor (or equivalently, the only right zero divisor).
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.
In mathematics, the term linear function refers to two distinct but related notions: [1] In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. [2] For distinguishing such a linear function from the other concept, the term affine function is often used ...
A divisor of that is not a trivial divisor is known as a non-trivial divisor (or strict divisor [6]). A nonzero integer with at least one non-trivial divisor is known as a composite number , while the units −1 and 1 and prime numbers have no non-trivial divisors.