Properties
Clearly, if R is a commutative ring, then so is R/I; the converse however is not true in general.
The natural quotient map p has I as its kernel; since the kernel of every ring homomorphism is a two-sided ideal, we can state that two-sided ideals are precisely the kernels of ring homomorphisms.
The intimate relationship between ring homomorphisms, kernels and quotient rings can be summarized as follows: the ring homomorphisms defined on R/I are essentially the same as the ring homomorphisms defined on R that vanish (i.e. are zero) on I. More precisely: given a two-sided ideal I in R and a ring homomorphism f : R → S whose kernel contains I, then there exists precisely one ring homomorphism g : R/I → S with gp = f (where p is the natural quotient map). The map g here is given by the well-defined rule g = f(a) for all a in R. Indeed, this universal property can be used to define quotient rings and their natural quotient maps.
As a consequence of the above, one obtains the fundamental statement: every ring homomorphism f : R → S induces a ring isomorphism between the quotient ring R/ker(f) and the image im(f). (See also: fundamental theorem on homomorphisms.)
The ideals of R and R/I are closely related: the natural quotient map provides a bijection between the two-sided ideals of R that contain I and the two-sided ideals of R/I (the same is true for left and for right ideals). This relationship between two-sided ideal extends to a relationship between the corresponding quotient rings: if M is a two-sided ideal in R that contains I, and we write M/I for the corresponding ideal in R/I (i.e. M/I = p(M)), the quotient rings R/M and (R/I)/(M/I) are naturally isomorphic via the (well-defined!) mapping a + M ↦ (a+I) + M/I.
In commutative algebra and algebraic geometry, the following statement is often used: If R ≠ {0} is a commutative ring and I is a maximal ideal, then the quotient ring R/I is a field; if I is only a prime ideal, then R/I is only an integral domain. A number of similar statements relate properties of the ideal I to properties of the quotient ring R/I.
The Chinese remainder theorem states that, if the ideal I is the intersection (or equivalently, the product) of pairwise coprime ideals I1,...,Ik, then the quotient ring R/I is isomorphic to the product of the quotient rings R/Ip, p=1,...,k.
Read more about this topic: Quotient Ring
Famous quotes containing the word properties:
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (16321704)
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—Ralph Waldo Emerson (18031882)