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An element that is a left or a right zero divisor is simply called a zero divisor. [2] An element a that is both a left and a right zero divisor is called a two-sided zero divisor (the nonzero x such that ax = 0 may be different from the nonzero y such that ya = 0). If the ring is commutative, then the left and right zero divisors are the same.
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. In field theory, the expression is only shorthand for the formal expression ab −1, where b −1 is the multiplicative inverse of b.
There are infinitely many other non-isomorphic commutative, non-associative, finite-dimensional real divisional algebras, but they all have dimension 2. In fact, every finite-dimensional real commutative division algebra is either 1- or 2-dimensional. This is known as Hopf's theorem, and was proved in 1940. The proof uses methods from topology.
For example, there are six divisors of 4; they are 1, 2, 4, −1, −2, and −4, but only the positive ones (1, 2, and 4) would usually be mentioned. 1 and −1 divide (are divisors of) every integer. Every integer (and its negation) is a divisor of itself. Integers divisible by 2 are called even, and integers not divisible by 2 are called odd.
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 . Elements a and b of an integral domain are associates if both a ∣ b {\displaystyle a\mid b} and b ∣ a {\displaystyle b\mid a} .
A prime divisor or irreducible divisor on X is an integral closed subscheme Z of codimension 1 in X. A Weil divisor on X is a formal sum over the prime divisors Z of X, , where the collection {:} is locally finite.
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Because of the identity (a + b) 2 − a 2 − b 2 = ab + ba, it follows that B(a, b) is real. Furthermore, since a 2 ≤ 0, we have: B(a, a) > 0 for a ≠ 0. Thus B is a positive-definite symmetric bilinear form, in other words, an inner product on V. Let W be a subspace of V that generates D as an algebra and which is minimal with respect to ...