Rings

In linear algebra in the last article we learned about vectors and vector spaces. We could add vectors and multiply them by scalars. But what if we have certains objects which themselves can be added and multiplied ? Rings provide us a framework and structure to study such objects. For example rings can contain a set of integers, polynomial or even matrices.

Formal Definition of Ring

A ring $\mathbb R$ is a set with two laws composition $+$ and $×$, that satisfy these axioms :

• Law of composition $+$ $\mathbb R$ is an Abelian group that we denote by $\mathbb R^+$, its identity denoted by $0$.
• Multiplication is associative.
• There exists a multiplicative identity denoted by 1.
• Multiplication distributes over addition.
• Distributive law $$(a + b)c = ab + bc, \forall \space a,b,c \in \mathbb R$$

In this article we will only consider commutative rings with identity since these are the rings used in coding theory.

A Subring is a subset that is itself a ring under the inherited operations.

A subset $S \subseteq \mathbb{R}$ is called a subring if

• $0 \in S$,
• closed under addition: $$a, b \in S \implies a + b \in S,$$
• closed under additive inverses: $$a \in S \implies -a \in S,$$
• closed under multiplication: $$a, b \in S \implies ab \in S,$$
• contains the multiplicative identity: $$1 \in S.$$

An Abelian group is a group in which the order of the operation does not matter; that is, for any two elements $a$ and $b$, $a + b = b + a$ or we can say that the group operation is commutative.

Polynomial Rings

A polynomial with coefficients in a ring $\mathbb R$ os a finite linear combination of powers of the variables.

here $a_i \in \mathbb R$. $f(x)$ is called formal polynomial. Set of polynomials with coefficient in ring $\mathbb R$ will be denoted by $\mathbb R[x]$.

The monomals $x^i$ are considered independent, So if :

$$g(x) = b_mx^m + b_{m-1}x^{m-1} + ... + b_1x + b_0$$

is another poly with coefficient in $\mathbb R$ then $f(x) == g(x)$ if and only if

$$a_i = b_i, \space \forall \space i = 0, 1, 2 ....$$

Now before introducing Quotient Rings which helps us to actually understand how extensions work we should go over some topics which will help us understand the construction of quotient rings.

Concepts

Ideal

$I$ is an ideal if :

• $I$ is an additive subgroup of $\mathbb R$. That means
  • $0 \in I$.
  • closed under addition.
  • closed under additive inverse.
• It absorbs multiplication : $$\forall \space r \in \mathbb R, a \in I$$ $$ra, ar \in I$$

Cosets

Let $I$ be an ideal of a ring $\mathbb{R}$. Since every ideal is an additive subgroup of $\mathbb{R}$ we can partition the ring into cosets of $I$.

For any element $a \in \mathbb{R}$, the coset of $I$ containing $a$ is

In other words, we take every element of the ideal and add $a$ to it.

For Example,

let

then

similarly

Every element of the ring belongs to exactly one coset and together the cosets partition the ring.

Quotient Set

The quotient set of a ring $\mathbb{R}$ by an ideal $I$ is the set of all cosets of $I$. It is denoted by

Here the elements of $\mathbb{R}/I$ are not individual elements of the ring. Instead, each element is an entire coset.

For Example,

let

then the quotient set is

Since every integer leaves one of the remainders $0, 1, 2,$ or $3$ when divided by $4$, these are the only distinct cosets. For convenience we will write

where

denotes the coset containing $a$.

The quotient set itself is only a collection of cosets. In the next section we will define addition and multiplication on these cosets, turning the quotient set into a quotient ring.

Ring Homomorphism

A ring homomorphism is a function between two rings that preserves the ring operations. In simple words performing addition or multiplication before applying the function gives the same result as applying the function first and then performing the operation.

Formally, let $\mathbb{R}$ and $\mathbb{R}'$ be two rings. A function

is called a ring homomorphism if, for every $a, b \in \mathbb{R}$,

• it preserves addition:

  $$\varphi(a + b) = \varphi(a) + \varphi(b),$$

• it preserves multiplication:

  $$\varphi(ab) = \varphi(a)\varphi(b),$$

• and it preserves the multiplicative identity:

  $$\varphi(1_{\mathbb{R}}) = 1_{\mathbb{R}'}$$

For Example,

Consider the function

defined by

This map sends every integer to its corresponding coset modulo $4$. For example,

It preserves both addition and multiplication, making it a ring homomorphism.

Kernel of a Ring Homomorphism

Let

be a ring homomorphism. The kernel of $\varphi$ is the set of all elements in $\mathbb{R}$ that are mapped to the additive identity of $\mathbb{R}'$. It is denoted by :

The kernel measures which elements of the ring become zero under the homomorphism. It is always an ideal of $\mathbb{R}$.

For Example,

Consider the ring homomorphism

defined by

An integer is mapped to the zero coset if and only if it is a multiple of $4$. Therefore,

Thus, every multiple of $4$ is identified with $0$ in the quotient ring $\mathbb{Z}/4\mathbb{Z}$.

Quotient Rings

We have already seen that an ideal $I$ partitions a ring $\mathbb{R}$ into disjoint cosets and that the collection of all these cosets forms the quotient set

The next step is to turn this quotient set into a ring by defining addition and multiplication on its cosets.

Addition

Addition of two cosets is defined by adding their representatives:

This operation is well-defined because $I$ is an additive subgroup of $\mathbb{R}$.

Multiplication

Similarly, multiplication is defined by

At first glance this definition may seem ambiguous since each coset contains many representatives. For example,

where $u, v \in I$

We need to verify that choosing different representatives does not change the resulting coset.

Consider the product of two arbitrary representatives:

Since $I$ is an ideal,

Therefore,

which implies

Hence every possible product of representatives belongs to the same coset $ab + I$ making multiplication well-defined.

The Quotient Ring

With these operations, the quotient set becomes a ring called the quotient ring denoted by :

Its elements are the cosets of the ideal $I$ while addition and multiplication are performed using the representatives of those cosets.

Example

Consider the ring

The quotient ring is

Addition and multiplication are performed modulo $4$. For example,

and

A quotient ring can be viewed as a new ring obtained by treating every element of the ideal $I$ as equal to zero. This construction is one of the most powerful tools in algebra and forms the foundation for constructing field extensions such as

$$\mathbb{F}_2[x]/(f(x)),$$

where $f(x)$ is an irreducible polynomial.

Adjoining Elements

One of the most powerful applications of quotient rings is that they allow us to construct new rings by introducing elements that did not previously exist.

Suppose we have a ring $\mathbb{R}$ and we wish to introduce a new element $\alpha$ satisfying a particular polynomial equation

where $f(x) \in \mathbb{R}[x]$.

If such an element does not already exist in $\mathbb{R}$ we can build a larger ring containing it.

Step 1: Introduce a Formal Symbol

We begin with the polynomial ring

Here, $x$ is simply a formal symbol. It is not yet a number or an element satisfying any special property. The only rules it obeys are those required by the ring axioms.

Step 2: Impose a Relation

Suppose we want our new element to satisfy

To enforce this relation, we form the quotient ring

where $(f(x))$ denotes the ideal generated by the polynomial $f(x)$.

Since every polynomial in the ideal becomes equal to the zero element in the quotient ring, we have

inside the quotient.

If we denote the residue of $x$ in the quotient ring by

then

exactly as desired.

In this way, we have adjoined a new element satisfying the required polynomial relation.

Example: Constructing the Complex Numbers

The real numbers $\mathbb{R}$ do not contain an element whose square is $-1$. We therefore seek to construct a larger ring containing an element $i$ satisfying

Rewriting this equation,

Now we want $i$ to be root of the polynomial :

We begin with the polynomial ring

and form the quotient ring

Inside this quotient ring,

which implies

The residue class of $x$,

is denoted by $i$

Therefore,

Every polynomial in $x$ can now be simplified using the relation

For example,

Hence every element of the quotient ring can be written uniquely in the form

Renaming the residue class of $x$ as $i$, every element becomes

which is precisely the familiar form of a complex number.

Thus,

More generally, adjoining a root of a polynomial $f(x)$ is achieved by forming the quotient ring

$$\mathbb{R}[x]/(f(x))$$

This construction is fundamental in algebra and is exactly how finite fields such as

$$\mathrm{GF}(2^m) \cong \mathbb{F}_2[x]/(f(x))$$

are constructed, where $f(x)$ is an irreducible polynomial of degree $m$.