Perhaps the simplest way is with an analogy: ideals are to rings as normal groups are to

groups.

Indeed, when you think of normal groups you may think of cosets, factor groups, and kernels

of homomorphisms.

And these concepts do carry over to ideals in rings.

Let’s see how…

To start, let’s recall the key properties of normal subgroups.

We’ll start with a group G.

A normal subgroup is a subgroup N of G… that divides G into “cosets”.

You can treat the cosets as elements in a new group, which is called a “factor group.”

It’s also called a “quotient group.”

But this doesn’t happen with every subgroup of G.

It only happens when “g” times “N” times “g-inverse” equals N for every single

element in the group G.

If this is true, then N is called a normal subgroup of G and we show this with a little

triangle.

Quick note: A regular subgroup that’s not normal is written with a less-than-or-equal

sign.

Let’s see some examples of normal subgroups.

Our group will be the group of integers with addition as the group operation.

Since this group is abelian, every subgroup is also a normal subgroup.

The subgroups are the multiples of 0..

1..

2…

3…

4… etc.

The first two subgroups are the group of just zero, and the entire set of integers.

Every group has at least these two normal subgroups: the group of just the identity

element, and the entire group.

But these normal subgroups do not tell us anything about the group.

Let’s look more closely at the multiples of 6.

This is a normal subgroup of the integers, so we write it like this…

This normal subgroup partitions the integers into 6 cosets, but only the normal subgroup

is a group.

The other 5 cosets are simply sets.

If we write the 6 cosets like this... then the factor group is a finite group with 6

elements.

This is the abstract algebra way of talking about the integers mod 6.

The cosets are the congruence classes, and we write “A is congruent to B mod 6” whenever

A and B are in the same congruence class.

So a normal subgroup N partitions a group G into cosets.

And if we treat the cosets as elements, then they also form a group which we call a “factor

group” or a “quotient group.”

The factor group is a completely separate group from N and G.

The name “factor group” conjures up images of factoring numbers.

This was deliberate.

Normal subgroups and factor groups are used to decompose groups into simpler parts.

Much like how the integers can be factored into primes, and molecules can be broken apart

into atoms, finite groups can be reduced to simple groups.

We will now take these ideas and generalize them to rings.

A ring R is a structure with two operations: addition and multiplication.

The ring is an abelian group under addition.

Multiplication, however, is a different story.

Multiplication is associative and closed.

But it may or may not be commutative.

And elements in the ring may or may not have inverses.

Also, there’s still some debate about whether a ring should be required to have an identity

element 1 for multiplication.

If “R” does have a multiplicative identity, we’ll call it a “ring with identity”

to make things clear.

And finally, addition and multiplication are linked by the distributive properties.

Suppose we have a ring R, and we want some subset “I” to generate cosets that will

act like a ring.

Display text: “I” for “Ideal” What properties must “I” have?

For starters, since R is an abelian group under addition, we definitely want “I”

to be a normal subgroup of “R” under addition.

And because R is abelian, every subgroup is normal.

So this gives us our first requirement: “I” must be an additive subgroup of “R”.

Since “I” is a subgroup, we can cover “R” in its cosets…

There may be a finite or infinite number of cosets.

Let’s pick two cosets at random: “x + I” and “y + I”.

Because “I” is a normal subgroup of R, if we add these two cosets together we get

the coset (x+y) + I.

But R is a ring, so we want to be able to multiply two cosets.

If we multiply these two cosets, we would hope to get “XY plus I.”

For this to be true, then if we pick ANY element

in the first coset… and multiply it by ANY element in the second coset… we should get

an element in the third coset…

If you expand the left hand side...

And then cancel the “XY” terms... you

get this:

In other words, no matter which two elements we pick from the cosets, the expression on

the left hand side should always be an element of I.

But it’s not clear WHY this should be true.

However, there’s a small trick we can use to find out when this does happen.

First, pick ANY element in the coset “X + I”, but from the second coset we’ll

pick just Y…

The element Y is in the coset “Y + I” because it’s equal to Y + 0.

Now, we want the product of these elements to be in the coset XY + I.

So if we multiply the elements on the left, we should get XY + i-sub-2 for some i2 in

the set I.

Multiplying out and cancelling the XY terms shows that I1 times Y equals I2.

That is, for any element Y in the ring R, I1 times Y is also in I for any element I-sub-1.

We can add this to our list of required properties for an ideal I.

We’ll just drop the index when adding it to our list of requirements.

Next, let’s pick the element X from the coset X + I, and then pick ANY element Y +

I1 from the second coset.

Like before, the product of these two elements should be in the coset XY + I.

This means X times (Y + I1) should equal XY + I2 for some element I2 in the ideal.

Once more we multiply out and cancel the XY terms, and find that X-times-I1 equals I-sub-2,

which means that X-times-I1 must be in the set I.

So for any element in the ring X, X-times-I1 must also be in I.

Let’s add this to our list of requirements, too.

Believe it or not, this is enough.

If we return to our earlier work, we wanted to show that this expression was an element

of the ideal.

We have just shown for coset multiplication to work, the first and second terms must be

in I. Visual: i1y + xi2 + i1i2 = i3

If we subtract these terms from both sides we get this.

On the right hand side, all three terms are in I.

And since I is an abelian subgroup, this difference is also in I.

Additionally this shows that the product of any two elements in I is also in I.

That is, “I” is closed under multiplication.

So we are actually done building our definition of an ideal I.

If these requirements are met, then the cosets of R can be added AND multiplied together.

We call “I” an ideal, and if we treat the cosets as new elements, they form a ring.

This is called a “quotient ring” or a “factor ring.”

So an ideal “I” of a ring R is a subgroup of R with two additional properties: for any

element R in the ring, and X in the ideal, both XR and RX are in the ideal.

These additional properties are a generalization of the concept of “multiples”.

For example, if you multiply any integer by a multiple of 5, you get another multiple

of 5.

And with ideals, if you multiply an element in the ring by an element in the ideal, you

get another element in the ideal.

Quick note.

The ideal is a normal subgroup that’s also closed under multiplication.

It inherits all the useful rules of arithmetic from the ring R. So “I” is ALMOST a subring.

What’s the one thing that’s missing?

The ONE thing?

That would be the element “one”…

The ONE thing that we’re missing here at Socratica is YOU.

We love making these advanced math videos for you, but we could use your help.

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Your help means we can continue making our videos, free for the world.

Now THAT would be IDEAL.

Let’s see an example of an ideal.

For our ring, we’ll use the polynomials with integer coefficients.

And for the ideal, let’s look at the multiples of “x”.

Visual: x*Z[x] = J We’ll call this set “J”.

This is the set of all polynomials with a constant term of 0.

Question: Does this satisfy the requirements to be an ideal?

Let’s check.

We first need to show that “J” is a subgroup of R.

The associative and commutative properties are inherited from the ring.

It has an identity element, since 0 is a polynomial with constant term 0.

When you add two polynomials, you add like terms.

And since both polynomials have a constant term of 0, their sum will, too.

So “J” is closed under addition.

And the additive inverse of a polynomial in “J” is also in “J”, because the inverse

of a polynomial with a constant term of 0, is also a polynomial with constant term 0.

Next, we need to show that for any polynomial F(X) in the ring and J(X) in J, both F(x)-times-J(x)

and J(x)-times-F(x) are in J. Because the ring is commutative, these two

are equal.

So we’ll just check that F(x)-times-J(x) is in “J”

Since J(X) is in J, it has a constant term of 0.

This means we can factor J(X) as X-times-K(X) for some other polynomial K.

Substituting this in and rearranging, we get X-times-F(x)-times-K(x).

And this is in J, because by definition, X times ANY polynomial in our ring is in J.

So the set “J” IS an ideal.

When an ideal is generated by a single element, we call it a “principal ideal” and write

it like this.

For instance, in the example we just saw, the multiples of “x” form a principal

ideal.

And for some rings, EVERY ideal is a principal ideal.

We call these rings “principal ideal domains”, or PIDs for short.

An ideal of a ring R is a subgroup I with two additional properties:

For any ‘r’ in R, and ‘x’ in I, then both “r times x” and “x times r” are

in the ideal I.

Sometimes you will encounter a subgroup I that satisfies one or the other of these properties,

but not both.

If only the first property is met, we call I a “left ideal”

And if only the second property is met, then we call I a “right ideal”

So if you are feeling somewhat daring, you could say that an ideal is a subgroup I that

is both a left- and right-ideal.

An ideal of a ring lets you treat the cosets as elements in a new ring.

As with groups, we call the collection of cosets a “factor ring” or a “quotient

ring.”

With such strong similarities you may ask yourself: “why not call “I” a normal

subring instead of an ideal?

The reason is historical.

The concept of an ideal came about in the late 19th century when number theorists were

working to generalize the ideas of integers, prime numbers, and factoring.

But there’s another reason.

Ideals are not technically subrings because they do not have a multiplicative identity.

Ooooh, disqualified.

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