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:

    “The most distinct and beautiful statement of any truth must take at last the mathematical form.”
    —Henry David Thoreau (1817–1862)

    “For me, the principal fact of life is the free mind. For good and evil, man is a free creative spirit. This produces the very queer world we live in, a world in continuous creation and therefore continuous change and insecurity. A perpetually new and lively world, but a dangerous one, full of tragedy and injustice. A world in everlasting conflict between the new idea and the old allegiances, new arts and new inventions against the old establishment.”
    —Joyce Cary (1888–1957)

    “It is well worth the efforts of a lifetime to have attained knowledge which justifies an attack on the root of all evil—viz. the deadly atheism which asserts that because forms of evil have always existed in society, therefore they must always exist; and that the attainment of a high ideal is a hopeless chimera.”
    —Elizabeth Blackwell (1821–1910)

    “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)