Abstract Algebra/Rings, ideals, ring homomorphisms

Basic definitions 0 100% #FFFFFF Definition 10.1: A ring is a set together with two binary operations and and two special elements, the unit and the zero , such that: is an abelian group with respect to with neutral element . is a monoid (that is, a group without inversion) with respect to with neutral element . The distributive laws hold: , . Examples 10.2: The whole numbers with respect to usual addition and multiplication are a ring. Every field is a ring. If is a ring, then all polynomials over form a ring. This example will be explained later in the section on polynomial rings. 0 100% #FFFFFF Definition 10.3: Let be a ring. A left ideal of is a subset such that the following two things hold: is a subgroup of . , where (closedness by left multiplication). Replacing closedness by left multiplication by closedness by right multiplication, we can define right ideals, and then both-sided ideals. If is a both-sided ideal of , we write . We'll now show an important property of the set of all ideals of a given ring, namely that it's inductive. This means: 0 100% #FFFFFF Definition 10.4: Let be a partially ordered set (that is, the usual conditions transitivity, reflexivity and anti-symmetry are satisfied). is called inductive if and only if every ascending chain of elements of (that is, a sequence in such that ) has an upper bound (that is, an element such that ). With this definition, we observe: 0 100% #FFFFFF Theorem 10.5: If a commutative ring is given, the set of all ideals of , partially ordered by inclusion (i.e. , where we use the convention of Donald Knuth and denote the power set of a set by ) is inductive. Proof: If is an ascending chain of ideals, we set and claim that . Indeed, if , find such that and . Then set , so that since . Similarly, if and , pick such that , whence since . Residue class rings 0 100% #FFFFFF Definition and theorem 10.4: Let be a ring, and . Then we define a relation on as follows: . This relation is an equivalence relation, and an equivalence class shall be denoted by for . If we define an addition and a multiplication , then these two are well-defined (i. e. independent of the choice of the representatives and ) and turn into a ring, called the residue class ring with respect to the ideal . Proof: First, we check that is an equivalence relation. Reflexiveness: since is an additive subgroup. Symmetry: since inverses are in the subgroup. Transitivity: Let and . Then , since a subgroup is closed under the group operation. Then we check that addition and multiplication are well-defined. Let and . Then for certain . Furthermore, for these same ; this is in by closedness by left and right multiplication. The ring axioms directly carry over from the old ring . Ring homomorphisms 0 100% #FFFFFF Definition 10.5: Let be rings. A ring homomorphism between the two is a map such that: For all and . ( is the unit of and of ).