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:
“Eroticism has its own moral justification because it says that pleasure is enough for me; it is a statement of the individual’s sovereignty.”
—Mario Vargas Llosa (b. 1936)
“Can a woman become a genius of the first class? Nobody can know unless women in general shall have equal opportunity with men in education, in vocational choice, and in social welcome of their best intellectual work for a number of generations.”
—Anna Garlin Spencer (1851–1931)
“Ye say they all have passed away,
That noble race and brave;
That their light canoes have vanished
From off the crested wave;
That, mid the forests where they roamed,
There rings no hunters’ shout;
But their name is on your waters,
Ye may not wash it out.”
—Lydia Huntley Sigourney (1791–1865)