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In mathematics, the additive identity of a set that is equipped with the operation of addition is an element which, when added to any element x in the set, yields x.One of the most familiar additive identities is the number 0 from elementary mathematics, but additive identities occur in other mathematical structures where addition is defined, such as in groups and rings.
The base case b = 0 follows immediately from the identity element property (0 is an additive identity), which has been proved above: a + 0 = a = 0 + a. Next we will prove the base case b = 1, that 1 commutes with everything, i.e. for all natural numbers a, we have a + 1 = 1 + a.
An identity with respect to addition is called an additive identity (often denoted as 0) and an identity with respect to multiplication is called a multiplicative identity (often denoted as 1). [3] These need not be ordinary addition and multiplication—as the underlying operation could be rather arbitrary.
These conditions imply that additive inverses and the additive identity are preserved too. If in addition f is a bijection, then its inverse f −1 is also a ring homomorphism. In this case, f is called a ring isomorphism, and the rings R and S are called isomorphic. From the standpoint of ring theory, isomorphic rings have exactly the same ...
In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest positive number of copies of the ring's multiplicative identity (1) that will sum to the additive identity (0). If no such number exists, the ring is said to have characteristic zero.
The zero ring is the unique ring in which the additive identity 0 and multiplicative identity 1 coincide. [1] [6] (Proof: If 1 = 0 in a ring R, then for all r in R, we have r = 1r = 0r = 0. The proof of the last equality is found here.) The zero ring is commutative. The element 0 in the zero ring is a unit, serving as its own multiplicative ...
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In mathematics, an empty sum, or nullary sum, [1] is a summation where the number of terms is zero. The natural way to extend non-empty sums [2] is to let the empty sum be the additive identity.