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:

    “He has the common feeling of his profession. He enjoys a statement twice as much if it appears in fine print, and anything that turns up in a footnote ... takes on the character of divine revelation.”
    —Margaret Halsey (b. 1910)

    “In our country today, very few children are raised to believe that their principal destiny is to serve their family, their country, or God.”
    —Benjamin Spock (b. 1903)

    “blind wantons like the gulls who scream
    And rip the edge off any ideal or dream.”
    —Louis MacNeice (1907–1963)

    “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)