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
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:
“If we do take statements to be the primary bearers of truth, there seems to be a very simple answer to the question, what is it for them to be true: for a statement to be true is for things to be as they are stated to be.”
—J.L. (John Langshaw)
“Silence, indifference and inaction were Hitlers principal allies.”
—Immanuel, Baron Jakobovits (b. 1921)
“As to the family, I have never understood how that fits in with the other idealsor, indeed, why it should be an ideal at all. A group of closely related persons living under one roof; it is a convenience, often a necessity, sometimes a pleasure, sometimes the reverse; but who first exalted it as admirable, an almost religious ideal?”
—Rose Macaulay (18811958)
“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 (18171862)
