Rings and Ideals

In the study of algebra, ring theory plays an important role in understanding the structural properties that arise in various algebraic systems. This field explores rings, which are sets equipped with two binary operations, and their special subsets called ideals. Our goal in this lesson is to present a comprehensive and plain-English explanation of these fundamental concepts that is both thorough and wide-ranging in their application.

What is a ring?

The ring (R, +, *) is a fascinating and essential algebraic structure. It is equipped with two operations we usually call addition and multiplication. More formally, a ring consists of a set R combined with two binary operations, addition + and multiplication *, where (R, +) is an abelian group, and (R, *) is a monoid.

Let us formalize the definition of a ring:

R is a set closed under two binary operations + and * such that for all a, b and c in R:
1. (R, +) is an abelian group.
   a. a + (b + c) = (a + b) + c (associativity of addition)
   b. There exists an element 0 in R such that a + 0 = a for every a in R (existence of additive identity)
   c. For every a in R, there exists an element -a such that a + (-a) = 0 (existence of additive inverses)
   d. a + b = b + a (commutativity of addition)

2. (R, *) is a semigroup.
   a. a * (b * c) = (a * b) * c (associativity of multiplication)

3. Distribution rules:
   A. a * (b + c) = (a * b) + (a * c)
   B. (a + b) * c = (a * c) + (b * c)

Example: The set of integers ℤ with simple addition and multiplication is a ring.

Rings do not have to have a multiplicative identity or be commutative (where a * b = b * a for every a and b), although certain types of rings do have these properties. When a ring is commutative and has a multiplicative identity, we call it a commutative ring with unity.

Visual example of a ring structure

Let's consider a visual example with a simple ring made of elements {0, 1, 2} under modulo-3 addition and multiplication:

Totals table:

    + | 0 1 2
    ,
    0 | 0 1 2
    1 | 1 2 0
    2 | 2 0 1

multiplication table:

    * | 0 1 2
    ,
    0 | 0 0 0
    1 | 0 1 2
    2 | 0 2 1

What is ideal?

An ideal is a special subgroup of a ring that plays an important role in ring theory, particularly in constructing quotient rings. Ideals generalize certain properties of numbers and functions and extend these concepts into the realm of ring theory.

Definition

Let R be a ring. A subset I of R is called a left ideal if:

1. (Additive closure) For any a, b in I, a + b is in I.
2. (Absorption property for left multiplication) For any r in R and a in I, r * a is in I.

Right ideals and two-sided ideals are defined similarly. A two-sided ideal, or simply an ideal, satisfies the absorption property for both left and right multiplication:

Right ideal: (additive closure) for any a, b in I, a + b is in I.
             (Absorption property for perfect multiplication) For any r in R and a in I, a * r is in I.

Bipartite ideal: (left ideal) For any r in R and a in I, r * a is in I.
                 (right ideal) For any r in R and a in I, a * r is in I.

Examples of ideals

Consider the ring of integers ℤ. An important example of an ideal in ℤ is the set of all multiples of an integer n, denoted by (n). For a specific integer n, the set is:

ℤ(n) = {kn : k ∈ ℤ}

This subset forms an ideal because:

• The sum of any two multiples of n will still be a multiple of n.
• For any integer m, the product of a multiple of m and n is also a multiple of n.

Forming a quotient ring using ideals

Quotient rings form the basis for defining equivalence relations within a ring. They allow us to "factor out" an ideal, thereby simplifying the structure of the ring. If I is an ideal of a ring R, then the quotient ring R/I is the set of cosets of I in R

Definition of quotient rings

The elements of R/I are of the form a + I, where a is in R, and arithmetic in R/I is defined by:

(a + i) + (b + i) = (a + b) + i
(a + i) * (b + i) = (a * b) + i

Given two elements a and b in R, the elements a + I and b + I are considered equivalent if their difference a - b is in the ideal I

Visual example of a quotient ring

Consider the ring of integers ℤ and the ideal (3) consisting of the multiples of 3:

(3) = {..., -3, 0, 3, 6, 9, ...}

The quotient ring ℤ/3ℤ has three equivalence classes:

0 + (3) = {..., -3, 0, 3, 6, 9, ...}
1 + (3) = {..., -2, 1, 4, 7, 10, ...}
2 + (3) = {..., -1, 2, 5, 8, 11, ...}

Each of the above classes contains integers which when divided by 3 give the same remainder.

Importance of ideals in ring theory

Ideals serve as building blocks for much of ring theory, just as normal subgroups do in group theory. They lead to the development of quotient rings and help characterize important ring properties such as prime and maximal ideals, which mirror concepts such as prime numbers in arithmetic and maximal subgroups in group theory.

Prime and maximal ideals

An ideal P within a commutative ring R is called prime if whenever a * b is in P, then either a is in P or b is in P

Maximal ideals are defined as follows: an ideal M in R is maximal if there are no other ideals between M and the whole ring R

Examples of prime and maximal ideals

In the integer ring ℤ, the ideal (p) is a prime ideal when p is a prime number (for example, (2), (3), (5) etc.). It is also maximal since there are no other integer ideals between (p) and ℤ.

Closing thoughts

Rings and ideals are the cornerstones of much of modern algebra. By understanding these concepts, mathematicians can explore the deeper and richer structures that define algebraic systems. Whether through the study of number systems, polynomials, or more abstract algebraic structures, the tools and terms described here unlock the frontiers of algebraic theory, guiding the development of further mathematical theory and applications.

Thus ring theory, while abstract, finds itself at the intersection of pure and applied mathematics, and reflects the symmetry and balance inherent in this mathematical structure.

Comments