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A zero morphism in a category is a generalised absorbing element under function composition: any morphism composed with a zero morphism gives a zero morphism. Specifically, if 0 XY : X → Y is the zero morphism among morphisms from X to Y , and f : A → X and g : Y → B are arbitrary morphisms, then g ∘ 0 XY = 0 XB and 0 XY ∘ f = 0 AY .
In algebra, the zero-product property states that the product of two nonzero elements is nonzero. In other words, =, = = This property is also known as the rule of zero product, the null factor law, the multiplication property of zero, the nonexistence of nontrivial zero divisors, or one of the two zero-factor properties. [1]
A zero element (or an absorbing/annihilating element) is an element z such that for all s in S, z • s = s • z = z. This notion can be refined to the notions of left zero, where one requires only that z • s = z, and right zero, where s • z = z. [2]
For any element x in a ring R, one has x0 = 0 = 0x (zero is an absorbing element with respect to multiplication) and (–1)x = –x. If 0 = 1 in a ring R (or more generally, 0 is a unit element), then R has only one element, and is called the zero ring. If a ring R contains the zero ring as a subring, then R itself is the zero ring. [6]
In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operation in question), just as the empty sum—the result of adding no numbers—is by convention zero, or the additive identity.
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]