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:

    “He has the common feeling of his profession. He enjoys a statement twice as much if it appears in fine print, and anything that turns up in a footnote ... takes on the character of divine revelation.”
    —Margaret Halsey (b. 1910)

    “General jackdaw culture, very little more than a collection of charming miscomprehensions, untargeted enthusiasms, and a general habit of skimming.”
    —William Bolitho (1890–1930)

    “We will have rings and things, and fine array,
    And kiss me, Kate, we will be married o’ Sunday.”
    —William Shakespeare (1564–1616)