Name: 環論の理想論 (抽象代数) (Ideals in Ring Theory (Abstract Algebra))
Uploaded: 2021-01-14T08:49:52.000Z
Duration: 11 min 57 s
Description: VoiceTubeの動画で発音を聞きながら英語表現を覚えよう！学べる英語：

There are many ways to motivate the concept of an “ideal” in abstract algebra.

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

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

And these concepts do carry over to ideals in rings.

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

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

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

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

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 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

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

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

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

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

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

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.

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

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

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

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

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

And then cancel the “XY” terms... you

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

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

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 +

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.

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環論の理想論 (抽象代数) (Ideals in Ring Theory (Abstract Algebra))

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