Quotient ring
Quotient ring [ edit ] Main article: Quotient ring The notion of quotient ring is analogous to the notion of a quotient group . Given a ring ( R , +, ⋅ ) and a two-sided ideal I of ( R , +, ⋅ ) , view I as subgroup of ( R , +) ; then the quotient ring R / I is the set of cosets of I together with the operations ( � + � ) + ( � + � ) = ( � + � ) + � , ( � + � ) ( � + � ) = ( � � ) + � . for all a, b in R . The ring R / I is also called a factor ring . As with a quotient group, there is a canonical homomorphism p : R → R / I , given by � ↦ � + � . It is surjective and satisfies the following universal property: If f : R → S is a ring homomorphism such that f ( I ) = 0 , then there is a unique homomorphism � ¯ : � /...