Chinese Remainder Theorem - Statement For Principal Ideal Domains

Statement For Principal Ideal Domains

For a principal ideal domain R the Chinese remainder theorem takes the following form: If u1, …, uk are elements of R which are pairwise coprime, and u denotes the product u1…uk, then the quotient ring R/uR and the product ring R/u1R× … × R/ukR are isomorphic via the isomorphism

such that

This map is well-defined and an isomorphism of rings; the inverse isomorphism can be constructed as follows. For each i, the elements ui and u/ui are coprime, and therefore there exist elements r and s in R with

Set ei = s u/ui. Then the inverse of f is the map

such that

g(a_1 + u_1R, \ldots, a_k + u_kR) = \left( \sum_{i=1}^k a_i \frac{u}{u_i} \left_{u_i} \right) + uR \quad\mbox{ for all }a_1, \ldots, a_k \in R.

This statement is a straightforward generalization of the above theorem about integer congruences: the ring Z of integers is a principal ideal domain, the surjectivity of the map f shows that every system of congruences of the form

can be solved for x, and the injectivity of the map f shows that all the solutions x are congruent modulo u.

Read more about this topic:  Chinese Remainder Theorem

Famous quotes containing the words statement, principal, ideal and/or domains:

    “Most personal correspondence of today consists of letters the first half of which are given over to an indexed statement of why the writer hasn’t written before, followed by one paragraph of small talk, with the remainder devoted to reasons why it is imperative that the letter be brought to a close.”
    —Robert Benchley (1889–1945)

    “Light, God’s eldest daughter, is a principal beauty in a building.”
    —Thomas Fuller (1608–1661)

    “The emancipation of today displays itself mainly in cigarettes and shorts. There is even a reaction from the ideal of an intellectual and emancipated womanhood, for which the pioneers toiled and suffered, to be seen in painted lips and nails, and the return of trailing skirts and other absurdities of dress which betoken the slave-woman’s intelligent companionship.”
    —Sylvia Pankhurst (1882–1960)

    “I shall be a benefactor if I conquer some realms from the night, if I report to the gazettes anything transpiring about us at that season worthy of their attention,—if I can show men that there is some beauty awake while they are asleep,—if I add to the domains of poetry.”
    —Henry David Thoreau (1817–1862)