Table of Contents

0. Preliminaries

Resources Used

Notation

1. Rings

From now on, fundamental knowledge of Group Theory (notes) are assumed.

Def. Ring

A set R with two binary operations + and \cdot, respectively called addition and multiplication, is called a ring if:

Notice R is necessarily non-empty as the additive identity 0 \in R.

A ring is said to be commutative (but not abelian) if the semigroup is commutative.

If the semigroup has an identity (that is, if the multiplication is a monoid) then its identity, denoted with 1 or 1_R, is called the identity element or the unity (of the ring). Such identity is always unique (exercise).

If the ring is with unity, then an element u \in R is said to be unit or invertible if there exists v \in R such that uv = vu = 1. Such v is unique and is called multiplicative inverse (or simply inverse) of u and is denoted with u^{-1}.

Do not mistake unit with unity. The unity is the trivial unit.

The set of all units in the ring R is denoted by \mathcal{U}(R).

The set of all non-zero elements of R is denoted by R^*.

The multiplication is called trivial if for all a,b \in R we have ab = 0.

Thm. Basic Ring Properties

Let R be a ring, then for all a,b \in R

  1. 0a = 0 = a0.
  2. -(a \cdot b) = (-a) b = a(-b).
  3. (-a)(-b) = ab

For all m,n \in \Z

  1. n(ab) = (na)b = a(nb).
  2. (mn)a = m(na) = n(ma).

Exercise

Thm. Basic Ring with Unity Properties

Let R be a ring with unity. Then

  1. The 0 is never an unit unless 0=1.
  2. 0=1 only if R=\{0=1\}, the trivial ring or the zero ring.
  3. If u and v are units in R, then so is uv and (uv)^{-1} = v^{-1}u^{-1}.
  4. \mathcal{U}(R) is a group under multiplication, called the group of units of R.

Def. Zero-Divisor

Let R be a ring and a \in R. Then a is called a left zero-divisor if there exists 0 \neq b \in R such that ab = 0. It is defined analogously for the right zero-divisior.

If a is either left or right zero-divisor, then it is said to be a zero divisor.

Def. Nilpotent Element

Let R be a ring and a \in R. Then a is said to be nilpotent if there exists an positive integer n such that a^n = 0.

Note that in any ring 0 is nilpotent which is called the trivial nilpotent element.

Def. Idempotent Element

Let R be a ring and a \in R. Then a is said to be idempotent if a^2 = a.

Similarly, in any ring 0 and, if exists, 1 are idempotent which are called trivial idempotents.

We say two idempotent elements are orthogonal to each other if ab = ba = 0.

Thm. (Binomial Theorem)

Let R be a ring with identity, n \in \N^+ and for a, b \in R we have ab=ba, then

(a+b)^n = \sum_{k=0}^n {{n \choose k} a^k b^{n-k}}

Def. Integral Domain

A non-zero ring R is called an integral domain if it has no non-trivial zero-divisors.

Therefore, let R be an integral domain, and a,b,c \in R. If ab = ac, then either a = 0 or b = c.

Def. Division Ring

A ring (R, +, \cdot) is called an division ring (or a skew-field) if, equivalently

  1. Every non-zero element of R, denoted R^*, has a multiplicative inverse, or
  2. (R^*, \cdot) forms a group.

Def. Field

A ring (R, +, \cdot) is called a field if, equivalently (exercise)

  1. It is a commutative division ring, or
  2. R^* is abelian under multiplication.
  3. It is a finite integral domain.

Thm. On Integral Domains, Division Rings, and Fields

Let R be a ring. Then

  1. If R is a field, then it is a division ring.
  2. If R is a division ring, then it is an integral domain.

Moreover,

  1. If R is a division ring, then multiplicative cancellation holds for non-zero elements.
  2. If R is an integral domain with unit, then only idempotent elements are 0 and 1.

Exercise

Thm. Basic Idempotent Properties

Let R be a ring, and a \in R idempotent. Then

  1. 1-a is idempotent as well.
  2. If a is non-trivial, it is a zero-divisor as well. This shows that integral domains and division rings do not have such idempotents.

Exercise

2. Subrings

Def. Subring

Let (R, +, \cdot) be a ring and S a non-empty subset of R. Then (S, +, \cdot) is called a subring if:

\{0\} and R are called the trivial subrings.

The center of R is, similar to groups, defined as

Z(R) = \{ r \in R \> | \> rx=xr \enspace \text{for all} \enspace x \in R \}

is a subring, and any subring of Z(R) is called a central subring.

Beware that existence of unity in subring or the ring does not imply existence of unity in the other. Indeed, if they both have unity, they are not necessarily equal.

Same issue is also true for the units. Remember, for multiplication operation, we are assumming sub-semigroup not subgroup.

Remarks

Let R be a ring and S \leq R its subring. Then

  1. S may be commutative even if R is not.
  2. S may not have unity even if R does. Moreover, S may have unity even if R does not.
  3. S and R may have different unities.

Exercise, show this is also the case for zero-divisors and units.

Def. Maximal Subring

Let R a ring and S a subring of R, then S is said to be maximal subring if S \neq R and for any subring T of R we have

S \subseteq T \subseteq R \implies T=S \enspace \lor \enspace T = R

Notice how we exclude the ring itself to be called maximal subring in itself.

Def. Opposite Ring

Given a ring R, the opposite ring is the ring with the same set of elements and same additive operation but multiplication reversed.

Thm. Self-Opposite iff Commutative

A ring R is it’s self-opposite (isomorphic to it’s opposite) if and only if R is commutative.

3. Ring Examples

Def. Ring of Continuous Functions

Let R be the set of real valued continious functions from the topological space X to \R. For f,g \in R, define the pointwise operations for all x \in X as

\begin{array}{rl} (f+g)(x) &= f(x) + g(x) \\ (fg)(x) &= f(x)g(x) \end{array}

Then R is a commutative ring with unity where the additive identity is the constant map \bold 0 and the unity is the constant map \bold 1.

Def. Matrix Ring

TODO

Def. Ring of Polynomials

Let R be a ring and X an indeterminate or variable over R. Define the set called ring of polynomials over R as

R[x] = \Set{ a_0 + a_1X + a_2X^2 + \cdots + a_nX^n | a_i \in R, n \in \N^+ }

We could have actually wrote n \in \N since X^0 = 1, but we don’t know if R is with identity.

then R[X] is a ring where addition and multiplication defined as expected over polynomials. Notice how elements of R[X] are of finite length, so this a set of finite polynomials.

Let a_0 + a_1X + \cdots + a_nX^n = p(X) \in R[X]. Then

Thm. Integral Domain Polynomials Properties

Let R be an integral domain and p(X), q(X) \in R[X], then

Proof


TODO:

Def. Power Series Ring

If we extend the definition of R[X] to infinite polynomials we have

R[[X]] = \Set{a_0 + a_1X + a_2X^2 + \cdots | a_i \in R}

which is called the power series over R and is also a ring (exercise).

Def. Laurent Ring and Series

Similar to ring of polynomials, Let R be a ring then a Laurent polynomial denoted with R[X^{\pm1}] or R[X, X^{-1}] is the set

\def\arraystretch{1.25} \begin{array}{ll} R[X^{\pm1}] = \{& a_{-n}X^{-n} + \cdots + a_{-1}X^{-1} + a_0 + a_1X \\ & + \cdots + a_mX^m \enspace | \enspace m,n \in \Z^*, \enspace a_i \in R\}. \end{array}

If we further extend this definition for the positive part as we did in power series the resulting set is called the Laurent series denoted with R\Braket{X}. Moreover, the bounded negative side is called the principal part and the other is called the power series part.

Def. Boolean Ring

A ring R in which every element is idempotent is called a boolean ring.

Thm. Structure Theorem for Boolean Rings

Every boolean ring is a subring of (\mathcal{P}(X), \triangle, \cap), the universal boolean ring, for some set X where the addition operation \triangle is the symmetric difference of two sets.

Notice that under these operations \mathcal{P}(X) is a commutative ring with unity. Moreover, a boolean ring need not to have unity. Take X infinite, then the subring which consists of all finite subsets of X has no unity.

Def. Group Rings

Let (R, +, \cdot) be a commutative ring with identity 1 \neq 0, and (G, *) = \Set{g_1, g_2, ..., g_n} a finite group. Define the group ring RG of G with coefficients in R as the set

RG = \Set{a_1g_1 + a_2g_2 + \cdots + a_ng_n | a_i \in R \enspace \text{and} \enspace 1 \leq i \leq n}

Notice a_ig_i multiplication is not defined.

If g_1 is the identity of G, then instead of a_1g_1, simply, a_1 will be written.

Addition and multiplication in RG is defined componentwise on coefficients canonically. This makes RG a ring (exercise).

Thm. Integers Modulo n

Let n \in Z^* so that \Z_n = \{\bar{0}, \bar{1}, ..., \overline{n-1}\}. Then \Z_n is a ring and for \bar{x} \in \Z_n

  1. \bar{x} is a unit \iff \bar{x} is not a zero-divisor \iff x is coprime to n.
  2. \Z_n is an integral domain \iff n prime \iff \Z_n is a field.
  3. \Z_n^* forms an additive subgroup \iff n is a power of a prime.
  4. If n prime, then x^n = x for all x \in \Z_n.

Exercise

Cartesian Two Product

Let \Set{R_i} be a non-empty family of rings, then \prod \Set{R_i} is a ring called the direct product of \Set{R_i}’s under component-wise addition and multiplication.

Remarks

Let T = \prod \Set{R_i} be the direct product of family of rings \Set{R_i} with at least two rings.

  1. T is a ring with identity if and only if each R_i is with identity.
  2. T is commutative if and only if each R_i is commutative.
  3. Even if each R_i is a field, T is not even an integral domain.

4. Ring Characteristic

Def. Characteristic

Let R be any ring. The characteristic of R, denoted by \text{Char}(R) is the least positive integer n such that na = 0 for all a \in R. If such n does not exists, then it is defined to be 0.

Thm. Basic Characteristic Properties

  1. \text{Char}(R) = 1 if and only if R = \Braket{0}.
  2. \text{Char}(R) = 0 if and only if the additive order |1| is infinite.
  3. \text{Char}(R) = n \neq 0 if and only if the additive order |1| is equal to n.

Example. Characteristic Examples

  1. \text{Char}(\Z) = \text{Char}(\mathbb{Q}) = \text{Char}(\R) = \text{Char}(\Complex) = \text{Char}(\mathbb{H}) = 0,
  2. \text{Char}(M_n(R)) = \text{Char}(R),
  3. \text{Char}(R) = \text{Char}(R[x]) = \text{Char}(R[[x]]),
  4. \text{Char}(\Z_n) = n
Proof


Exercise.

Thm. Characteristic of Cartesian Product Ring

Let R and S be rings, then their characteristic is

  1. 0 if either R or S has characteristic 0, or
  2. \text{lcm}(\text{Char}(R), \text{Char}(S)).

Thm. ’

Suppose R is a ring with 1 whose non-units forms a subgroup under addition. Then either,

  1. \text{Char}(R) = 0, or
  2. \text{Char}(R) = p^n where p prime and n positive integer.
Proof


TODO:

5. Ideals

Def. Ideal

Let R be a ring. A subset I of R is called a left (respectively right) ideal of R if

  1. I \leq (R, +), and
  2. for all a \in R we have aI \subseteq I (respectively Ia \subseteq I), under the ring multiplication.

If I is both a left and a right ideal, then it is called a two-sided ideal or a 2-sided ideal. Notice that in this case we have Ia = aI = I.

Noting a ring R is an ideal of itself, such ideal R is called the unit ideal. (0) = \{0\} is also an ideal in R called the zero ideal. These two ideals are called the trivial ideals of R.

Notice how the concept of an ideal is similar to the concept of a coset in group theory.

Notation. Ideal

Let R be a ring and x \in R, then

More on generations later.

Thm. On Improper Ideals

Let R be a ring and I a left (resp. right) ideal, then the following are equivalent

  1. I = R,
  2. 1 \in I,
  3. I has an unit, or just
  4. I has an element which has an left (resp. right) inverse.
Proof


Exercise.

Thm. Division Ring and Ideals

Let R be a ring with 1, then R is a division ring if and only if (0) and R are the only left ideals (or the only right ideals) in R.

This is also true if R, instead of a ring with identity, is a non-zero ring with non-trivial multiplication. For the proof of this theorem check out Musli pp. 43-44.

Proof


TODO:

Def. Maximal Ideal

A left (resp. right or 2-sided) ideal I of a ring R is called maximal ideal in R if for any left (resp. right or 2-sided) ideal J of R we have

I \subseteq J \subseteq R \implies J=I \enspace \lor \enspace J=R

where I \neq R. Thus, we exclude unit ideal to be called maximal ideal.

Def. Minimal Ideal

Similar to maximal ideal, a left (resp. right or 2-sided) ideal of R is called a minimal ideal in R if for any left (resp. right or 2-sided) ideal J of R we have

(0) \subseteq J \subseteq I \implies J = (0) \enspace \lor \enspace J=I

where I \neq (0). Thus, we exclude zero ideal to be called minimal ideal.

Thm. Existence of Maximal Ideal

Let R be a ring with 1 and I its left (resp. right or 2-sided) ideal such that I \neq R. Then there exists left (resp. right or 2-sided) maximal ideal M such that I \subseteq M.

This theorem need not to be true for minimal ideals even if the ring is commutative. For example, take the ring \Z and its ideal 2\Z.

Proof


TODO: Zorn’s Lemma and add partial order defn to preliminaries.

Def. Prime Ideal

Also see Wiki: Prime Ideal.

Let R be a commuative ring and I its ideal. I is called a prime ideal if I \neq R and for all x,y \in R

xy \in I \implies x \in I \enspace \lor \enspace y \in I.

Clearly non-trivial R is a commutative integral domain if and only if (0) is a prime ideal in R.

Thm. Nilpotents of a Commutative Ring

The set of all nilpotent elements in a commutative ring R with 1 is the intersection of all prime ideals.

Thm. Prime Avoidance Lemma

Let R be a commutative ring, A \leq R, and I_1, I_2, ..., I_n \trianglelefteq R such that I_i is prime for i \geq 3 (that is at most two ideals are not prime). Then

If A \not\subseteq I_j for any one j, then A \not\subseteq \bigcup_{1 \> \leq \> k \> \leq \> n} I_k. So that if A is not contained in any of the ideals, it is also not contained in their union.

6. Ideals and Generators

Def. Ideal Generator

Let R a ring and \subseteq R. Then the left ideal generated by X is the smallest left ideal in R which contain X, or equivalently intersection of all left ideals which contain X.

In particular, if X = \varnothing, then the left ideal generated by X is (0).

If X = \{x\}, then the left ideal generated by X is

\Set{nx + rx | n \in \Z, r \in R}

Notice that if the ring is with identity, then the set is equal to \Set{sx | s \in R}. This set is denoted by (x)_l called the left ideal generated by x.

Def. Principal Ideal

A left ideal I in R is called a principal left ideal if I=(x)_l for some x \in I.

Definition of right principal ideal is similar which is denoted by (x)_r.

(x) denotes the 2-sided ideal generated by x.

Def. Principal Ideal Ring

A ring R is called a principal ideal ring (P.I.R.) if

Def. Principal Ideal Domain

A principal ideal ring R which is an integral domain is called a principal ideal domain (P.I.D.).

Def. Finitely Generated Ideal

We say an ideal I in a ring R is finitely generated if there exists a finite X \subseteq R which generates I. Notice that in this case, every element of I can be expressed as a R-linear combinations of x_1,...,x_n \in X.

Thm. Basic Ideal Properties

Let R be a ring with 1 and I a left ideal of R, then the following are equivalent

  1. I = R,
  2. 1 \in I,
  3. I contains an unit, or in particular
  4. I contains an element which has a left inverse.

By symmetry, of course, this also holds for right ideals where (4) is right inverse.

For the 2-sided ideal case (4) is left inverse OR right inverse.

Corollary

Let I = (x)_l, then I=R if and only if x has a left inverse.

Thm. Ideals of Division Ring

Let R be a ring with 1, then the following are equivalent

Notice that if R is commutative ring with 1, then it is a field in this case.

This theorem also holds if R is a non-zero ring with non-trivial multiplication.

Ideals of Matrices

Let R be a ring with 1 and S = M_n(R), then if I is a left (resp. right or 2-sided) ideal in R, then M_n(I) is a left (right or 2-sided) ideal in S.

Simple Ring

A non-zero ring R is said to be a simple ring if only 2-sided ideals of R are (0) and R.

This implies that M_n(R) is simple if R is simple. Similarly, division rings are thefore simple.

7. Algebra of Ideals

Let R be a ring and I and J its ideals of the same kind. Then the addition of ideals I and J is defined as

I + J = \Set{i + j | i \in I, j \in J} \subseteq R

which is an ideal of the same kind in R. Such addition is commutative and associative.

The multiplication of ideals I and H is defined as

IJ = \Set{i_1j_1 + \cdots + i_nj_n | i_i \in I, j_i \in J}

that is the finite sums of products of pairs from I and J. This product is again an ideal in R. Such multiplication is associative but not necessarily commutative.

Notice how the multiplication in Ring Ideals differs from the subset/subring/coset multiplication in Group Theory.

Remark

Let R be a ring and I, J \subseteq R.

  1. IJ is a left ideal if I is an ideal.
  2. IJ is a right ideal if J is an ideal.
  3. IJ is a 2-sided ideal if I is a left ideal and J is a right ideal.
    1. IR \subseteq I and RI \subseteq I.
    2. Moreover, if R is with 1, then IR = I = RI.

Def. Nilpotent Ideal

Let I be an ideal. If for some n \geq 1 we have I^n = (0), then I is called a nilpotent ideal.

Def. Nil Ideal

Let I be an ideal such that every element of I is nilpotent, then I is called a nil ideal.

Thm. Nilpotent and Nil Ideal

Every nilpotent ideal I is a nil ideal. Converse is not necessarily true unless I is finitely generated.

8. Quotient Rings

Def. Quotient

Let R be a ring and I