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:
“Truth is used to vitalize a statement rather than devitalize it. Truth implies more than a simple statement of fact. “I don’t have any whisky,” may be a fact but it is not a truth.”
—William Burroughs (b. 1914)
“You have lived longer than I have and perhaps may have formed a different judgment on better grounds; but my observations do not enable me to say I think integrity the characteristic of wealth. In general I believe the decisions of the people, in a body, will be more honest and more disinterested than those of wealthy men.”
—Thomas Jefferson (1743–1826)
“We will have rings and things, and fine array,
And kiss me, Kate, we will be married o’ Sunday.”
—William Shakespeare (1564–1616)