Unique Factorization Domain - Definition

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:

    “According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
    —Ana Castillo (b. 1953)

    “Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
    —Elaine Heffner (20th century)

    “No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
    —Florence Nightingale (1820–1910)