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  • There are many ways to motivate the concept of anidealin abstract algebra.

  • 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.

  • Well start with a group G.

  • A normal subgroup is a subgroup N of G… that divides G intocosets”.

  • 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-inverseequals 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

  • groupor a “quotient group.”

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

  • The namefactor groupconjures 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, well 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” forIdealWhat 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 getXY plus I.”

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

  • in the first cosetand multiply it by ANY element in the second cosetwe should get

  • an element in the third coset

  • If you expand the left hand side...

  • And then cancel theXYterms... 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 well

  • 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.

  • Well 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 ringor 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 ofmultiples”.

  • 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 elementone”…

  • The ONE thing that were missing here at Socratica is YOU.

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

  • Did you know you can support Socratica on Patreon?

  • 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, well use the polynomials with integer coefficients.

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

  • Visual: x*Z[x] = J Well 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 well 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 idealand 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 ringsprincipal 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 ringor 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.

  • Support what you love! Join the Socratica Team on Patreon

There are many ways to motivate the concept of anidealin abstract algebra.

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B1 中級

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

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    林宜悉 に公開 2021 年 01 月 14 日
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