Definition
Formally, a unique factorization domain is defined to be an integral domain R in which every non-zero x of R can be written as a product (an empty product for the unit) of irreducible elements pi of R and a unit u:
- x = u p1 p2 ... pn with n≥0
and this representation is unique in the following sense: If q1,...,qm are irreducible elements of R and w is a unit such that
- x = w q1 q2 ... qm with m≥0,
then m = n and there exists a bijective map φ : {1,...,n} -> {1,...,m} such that pi is associated to qφ(i) for i ∈ {1, ..., n}.
The uniqueness part is usually hard to verify, which is why the following equivalent definition is useful:
- A unique factorization domain is an integral domain R in which every non-zero element can be written as a product of a unit and prime elements of R.
Read more about this topic: Unique Factorization Domain
Famous quotes containing the word definition:
“Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
—Walter Pater (1839–1894)
“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)
“I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)