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.

Definition 1. [Ring, Subring, Ring homomorphism]

A ring is a triple $(R, +: R \times R \rightarrow R, \times: R \times R \rightarrow R)$ such that:
1. $(R, +)$ is an abelian group;
2. For any $a, b, c \in R$, $\times(\times(a, b), c) = \times (a, \times (b, c))$;
3. For any $a, b, c \in R$:
  1. $\times(+(a, b), c) = \times(a, c) + \times(b, c)$;
  2. $\times(a, +(b, c)) = \times(a, b) + \times(a, c)$;
A ring $S \subseteq R$ is called subring if it's both closed under $+$ and $\times$.
Let $(R, +_R, \times_R)$, $(S, +_S, \times_S)$ be two rings. A map $f: R \rightarrow S$ is called a ring homomorphism if it's homomorphism for $(R, +) \rightarrow (S, +)$ and $f(a \times b) = f(a) \times f(b)$.

Remark 2.

If there is no ambiguity, we write $\times (a, b)$ as $a \times b$ or $ab$, $+(a, b)$ as $a + b$;
We write the identity of $(R, +)$ as 0;
The set of ring homomorphism from $R$ to $S$ is written as $Hom(R, S)$.

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

If $(R, \times)$ has identity, we say $R$ has identity, and $e_{(R, \times)}$ to be multiplicative identity. We usually write the identity of $(R, \times)$ as 1;
If $a \times b = b \times a$, we say $R$ is a communicative ring;
If $R$ has identity and is communicative, and $a \times b = 0$ implies one of $a$ and $b$ is 0, we call $R$ an integral domain.
If $(R\setminus \{0\}, \times)$ is an abelian group, we say $R$ is a field. It's clear that $R$ is a field iff it's an integral domain and $(R\setminus \{0\}, \times)$ has inverse.

This is the hardest part of ring theory in my opinion: U have to keep all of those concepts in ur mind :-(

Module

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

A (left) R-module is a triple $(M, +: M \times M \rightarrow M, \cdot: R \times M \rightarrow M)$, with $(M, +)$ is an abelian group, such that:
1. For any $a, b \in R$, $x \in M$, $(a \times b) \cdot x = a \cdot (b \cdot x)$;
2. For any $a, b, c \in R$:
  1. $(a + b) \cdot c = a \cdot c + b \cdot c$;
  2. $a \cdot (b + c) = (a \cdot b) + (a \cdot c)$;
A module $S \subseteq R$ is called submodule if it's both closed under $+$ and $\times$.
Let $(M, +_M, \cdot_M)$, $(N, +_N, \cdot_N)$ be two R-modules. A map $f: R \rightarrow S$ is called a R-module homomorphism if it's homomorphism for $(M, +_M) \rightarrow (N, +_N)$ and $f(a \cdot_M b) = f(a) \cdot_N f(b)$.

Remark 5. If $F$ is a field, $M$ an R-module, and for $\forall \; x \in M, 1\cdot x = x$, we call $M$ an $F$ vector space.

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

Let $R$ be a ring, $G$ a group. The group ring denoted as $R[G]$ consists of all maps from $R$ to $G$ such that finitely $g \in G$ is sent to non-zero values.
Let $Q = R \times (R \setminus \{0\}) / \sim$ with $(a, b) \sim (c, d)$ if $ac=bd$. Let $+_Q = ([a, b], [c, d]) \mapsto ([ad + bc, bd])$, $\times_Q = ([a, b], [c, d]) \mapsto ([ac, bd])$, then we have $Q$ is a field, called field of fractions.
When $K$ is field, we can show that $K[z]$ is an integral domain, then the field of fractions $K[z]$ is called field of rational functions, denote as $K(z)$.
When $K$ is a field, $K[x] = \{\sum_{i=1}^{n}a_ix^{i}\}$. Let $f=\sum_{i=1}^{n_f}f_ix^{i}$, we define the addition $(f+g) = \sum_{i=1}^{max(n_f, n_g)}(f_i+g_i)x^{i}$, with $f_i=0\;(i>n_f)$, $g_i=0\;(i>n_g)$, and the multiplication $(f+g) = \sum_{i=1}^{n_f + n_g}(\sum_{j=0}^{i}f_j g_{i - j})x^i$. 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 $R$ forms a group, denoted as $Aut(R)$.

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

A homomorphism $f$ is called R-module isomorphism if it is a bijection if it is between two R-modules.
The set of R-module automorphisms of $M$ forms a group, denoted as $Aut_R(M)$, which is a subgroup of permutation group $S_M$ (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 $(M, +_M)$ is an abelian group, every subgroup of $M$ is a normal subgroup.

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

M is a submodule;
The quotient group $M/N$ has a R-module structure, with scalar multiplication defined as $r \cdot (x + N) = rx + N$;
There is a R-module $Q$ and a R-module homomorphism $f: M \rightarrow Q$ such that $N = ker(f)$.

Here, the ring $Q$ is $M/N$, with $(x + N) + (y + N) = (x + y) + N$.

Theorem 11. Let $R$ be a ring, $I$ a subgroup of $(R, +)$, T.T.F.E.:

For any $a \in R$, $c \in I$, $a \times c \in I$ and $c \times a \in I$;
The quotient group $R / I$ is a ring with $(a + I) \times (b + I) = a \times b + I$;
There is a ring $S$ and a ring homomorphism $f: R \rightarrow S$, such that $I = ker(f)$.

Here, the ring $S$ is $R / I$, with $(a + I) + (b + I) = (a + b) + I$. 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 $R$ is a commutative ring with identity, $I$ an ideal of $R$, we have:

If $R / I$ is an integral domain, we call $I$ a prime ideal;
Moreover, if $R / I$ is a field, we call $I$ a maximal ideal.