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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.
An effective Cartier divisor on X is an ideal sheaf I which is invertible and such that for every point x in X, the stalk I x is principal. It is equivalent to require that around each x , there exists an open affine subset U = Spec A such that U ∩ D = Spec A / ( f ) , where f is a non-zero divisor in A .
One says that a is a two-sided divisor of b if it is both a left divisor and a right divisor of b; the x and y above are not required to be equal. 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 a ∣ b ...
The meaning of the expression should be the solution x of the equation =. 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.
Let X be a scheme which is finite type over a field k. An algebraic r-cycle on X is a formal linear combination [] of r-dimensional closed integral k-subschemes of X. The coefficient n i is the multiplicity of V i. The set of all r-cycles is the free abelian group
1. In an integral domain R, [clarification needed] an element a is called a divisor of the element b (and we say a divides b) if there exists an element x in R with ax = b. 2. An element r of R is a left zero divisor if there exists a nonzero element x in R such that rx = 0 and a right zero divisor or if there exists a nonzero element y in R ...
A divisor on a Riemann surface C is a formal sum = of points P on C with integer coefficients. One considers a divisor as a set of constraints on meromorphic functions in the function field of C, defining () as the vector space of functions having poles only at points of D with positive coefficient, at most as bad as the coefficient indicates, and having zeros at points of D with negative ...
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.