Chinese Remainder Theorem - Statement For General Rings

Statement For General Rings

The general form of the Chinese remainder theorem, which implies all the statements given above, can be formulated for commutative rings and ideals. If R is a commutative ring and I1, …, Ik are ideals of R which are pairwise coprime (meaning that for all ), then the product I of these ideals is equal to their intersection, and the quotient ring R/I is isomorphic to the product ring R/I1 × R/I2 × … × R/Ik via the isomorphism

such that

Here is a version of the theorem where R is not required to be commutative:

Let R be any ring with 1 (not necessarily commutative) and be pairwise coprime 2-sided ideals. Then the canonical R-module homomorphism is onto, with kernel . Hence, (as R-modules).

Read more about this topic:  Chinese Remainder Theorem

Famous quotes containing the words statement, general and/or rings:

    It is commonplace that a problem stated is well on its way to solution, for statement of the nature of a problem signifies that the underlying quality is being transformed into determinate distinctions of terms and relations or has become an object of articulate thought.
    John Dewey (1859–1952)

    Of what use, however, is a general certainty that an insect will not walk with his head hindmost, when what you need to know is the play of inward stimulus that sends him hither and thither in a network of possible paths?
    George Eliot [Mary Ann (or Marian)

    Ah, Christ, I love you rings to the wild sky
    And I must think a little of the past:
    When I was ten I told a stinking lie
    That got a black boy whipped....
    Allen Tate (1899–1979)