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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 .
the group under multiplication of the invertible elements of a field, [1] ring, or other structure for which one of its operations is referred to as multiplication. In the case of a field F, the group is (F ∖ {0}, •), where 0 refers to the zero element of F and the binary operation • is the field multiplication, the algebraic torus GL(1).
Zero is thus an absorbing element. The zero of any ring is also an absorbing element. For an element r of a ring R, r0 = r(0 + 0) = r0 + r0, so 0 = r0, as zero is the unique element a for which r − r = a for any r in the ring R. This property holds true also in a rng since multiplicative identity isn't required.
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]
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]
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]
unit or invertible element : An element r of the ring R is a unit if there exists an element r −1 such that rr −1 = r −1 r = 1. This element r −1 is uniquely determined by r and is called the multiplicative inverse of r. The set of units forms a group under multiplication. unity The term "unity" is another name for the multiplicative ...
A variation of this definition arises if S contains an absorbing zero that is different from the multiplicative unit e, it is required that the non-zero elements be invertible, and a·0 = 0·a = 0. Since multiplication is associative, the (non-zero) elements of a semifield form a group.