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:

    “Children should know there are limits to family finances or they will confuse “we can’t afford that” with “they don’t want me to have it.” The first statement is a realistic and objective assessment of a situation, while the other carries an emotional message.”
    —Jean Ross Peterson (20th century)

    “I would urge that the yeast of education is the idea of excellence, and the idea of excellence comprises as many forms as there are individuals, each of whom develops his own image of excellence. The school must have as one of its principal functions the nurturing of images of excellence.”
    —Jerome S. Bruner (20th century)

    “I’m no idealist to believe firmly in the integrity of our courts and in the jury system—that is no ideal to me, it is a living, working reality. Gentlemen, a court is no better than each man of you sitting before me on this jury. A court is only as sound as its jury, and a jury is only as sound as the men who make it up.”
    —Harper Lee (b. 1926)

    “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)