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:

    I think, therefore I am is the statement of an intellectual who underrates toothaches.
    Milan Kundera (b. 1929)

    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)

    You held my hand
    and were instant to explain
    the three rings of danger.
    Anne Sexton (1928–1974)