Ring

I took Abstract Algebra in UW-Madison(MATH541) with professor Chenxi Wu. This article is inspired by prof. Wu's lecture note, and many formats, content, and structure of this blog are influenced by and similar to it.

Ring

Definition 1. [Ring, Subring, Ring homomorphism]

• A ring is a triple such that:
  1. is an abelian group;
  2. For any , ;
  3. For any :
     1. ;
     2. ;
• A ring is called subring if it's both closed under and .
• Let , be two rings. A map is called a ring homomorphism if it's homomorphism for and .

Remark 2.

1. If there is no ambiguity, we write as or , as ;
2. We write the identity of as 0;
3. The set of ring homomorphism from to is written as .

Definition 3. [Identity, Communicative, Integral Domain, Field]

1. If has identity, we say has identity, and to be multiplicative identity. We usually write the identity of as 1;
2. If , we say is a communicative ring;
3. If has identity and is communicative, and implies one of and is 0, we call an integral domain.
4. If is an abelian group, we say is a field. It's clear that is a field iff it's an integral domain and has inverse.

This is the hardest part of ring theory in my opinion: U have to memorize all of these concepts!

Module

Definition 4. [R-module, submodule, module homomorphism]

• A (left) R-module is a triple , with is an abelian group, such that:
  1. For any , , ;
  2. For any :
     1. ;
     2. ;
• A module is called submodule if it's both closed under and .
• Let , be two R-modules. A map is called a R-module homomorphism if it's homomorphism for and .

Remark 5. If is a field, an R-module, and for , we call an vector space.

Definition 6. [Group ring, Field of fractions, Field of rational functions, Polynomial ring]

• Let be a ring, a group. The group ring denoted as consists of all maps from to such that finitely is sent to non-zero values.
• Let with if . Let , , then we have is a field, called field of fractions.
• When is field, we can show that is an integral domain, then the field of fractions is called field of rational functions, denote as .
• When is a field, . Let , we define the addition , with , , and the multiplication . We can show that this is a ring.

Definition 7. [Ring isomorphism / isomorphic, Group of ring automorphisms]

• A ring homomorphism is called ring isomorphism if it is bijection.
• The set of ring automorphisms of forms a group, denoted as .

Definition 8. [R-module isomorphism / isomorphic, Group of R-module automorphisms]

• A homomorphism is called R-module isomorphism if it is a bijection if it is between two R-modules.
• The set of R-module automorphisms of forms a group, denoted as , which is a subgroup of permutation group (Not all permutations are homomorphism!).

Isomorphism Theorem of Ring and Module

Definition 9. [Kernal, Image] The two definitions are almost exactly same as the ones of group.

Since is an abelian group, every subgroup of is a normal subgroup.

Theorem 10. Let be a ring, an R-module, is a subgroup (must be normal subgroup) of , T.T.F.E.:

1. M is a submodule;
2. The quotient group has a R-module structure, with scalar multiplication defined as ;
3. There is a R-module and a R-module homomorphism such that .

Here, the ring is , with .

Theorem 11. Let be a ring, a subgroup of , T.T.F.E.:

1. For any , , and ;
2. The quotient group is a ring with ;
3. There is a ring and a ring homomorphism , such that .

Here, the ring is , with . Such ring is called ideal (analogy of normal subgroups).

In ring theory, ideal plays the role of normal subgroups in group theory.

Definition 12. [Prime ideal, Maximal ideal] If is a commutative ring with identity, an ideal of , we have:

1. If is an integral domain, we call a prime ideal;
2. Moreover, if is a field, we call a maximal ideal.