Hello everyone, and welcome to Carmen's Core Concepts.
My name is Carmen Bruni,
and this is Week 7 Part 2.
So I felt the need to break off this last section because
it is fairly abstract that's for 1.
For 2, the other video's getting
quite long, so I thought I would cut it off there and start a new video here.
And thirdly, there's a lot of parts that I'm going to talk about in this video that are…
what do I want to say… sort of above and beyond what a
Math 135 student should know at the end of the course,
but at the same time, I feel like these concepts are very important
and that there's no reason not to include them
in Math 135.
So that being said,
those factors combined gave us a Week 7 Part 2 video
that is the following.
So wall of text, but
what am I going to say here? A ring is
a very common structure. You actually know what a ring is
without really knowing what a ring is.
The idea behind this slide and this definition is to give you...
a way to group
sets that you know together basically.
So let's describe what I mean by this.
So definition: “a commutative ring is a set R along with two closed operations,
addition and multiplication…”
by closed operations we mean if I add 2 elements of the set R,
I stay inside R, and if I multiply 2 elements of the set R, I stay inside R,
“…and these two operations satisfy for all
for all a, b, c inside R that they are...
that the operation is associative…” so both operations are associative.
So a plus b plus c is equal to a plus b plus c,
and same with multiplication. So here I've denoted - I’m not going to always use
the dot for multiplication. I'm just going to use juxtaposition
for multiplication so if there are two things right beside each other
that implies that they're multiplied together.
So a times b then times c is the same as a times b times c.
Commutative,
a plus b is the same as b plus a,
and a times b is the same as b [times] a, in a commutative ring.
So multiplication doesn't always have to be commutative in a ring,
but in a commutative ring it is.
We have identities; so there are distinct elements 0 and 1
such that a plus 0 is a, and a times 1 is a.
I've written “distinct” in brackets. Some books
don't require this, some books do.
Take your pick. It doesn't matter that much.
Additive Inverses, there exists an element minus a such that
a plus negative a is 0.
So every element inside your set
has an additive inverse.
And the distributive property, a times b plus c is equal to a times b plus a times c.
As I said at the beginning, well
a commutative ring is something that you already kind of know.
You already know examples of commutative rings:
the integers, the rational numbers the real numbers,
all of these are commutative rings.
The natural numbers don't form a commutative ring.
They are associative, they are commutative,
they don't really have an identity, right?
The additive identity isn't inside N. Remember our natural numbers start at 1.
And additive inverses are missing. This is the key one that's missing from the natural numbers.
There's no - if I have 1, I don't have negative 1 inside the natural numbers.
But I do inside the integers, and that's what forms a commutative ring.
Okay, so here's the definition. Again these are things that you know, right?
So I'm just kind of grouping together a set of objects that you
already know have these properties
and I'm giving them some sort of classification name, okay?
And I haven't defined a field yet. What's a field?
Maybe you can guess, but maybe you can’t.
So here we have additive inverses, a field adds multiplicative inverses.
So if, in addition, every non-zero element has a multiplicative inverse,
that is an element a inverse,
so we denote it by a to the power of negative 1,
and this element satisfies a times a inverse equals 1,
we say that R is a field.
Okay? So
every field is a ring,
every field is a commutative ring,
but not every ring is a field.
So for example, Q and R,
these are fields and these are commutative rings,
but the integers, for example, is not a field
because elements like 2, 3, 4
their inverses, or their multiplicative inverses,
1 over 2, 1 over 3, 1 over 4, those aren't inside the integers.
So that's why again so maybe this is giving you a little bit of context
as to why we have all these symbols and definitions and
how this all came about.
That's sort of what I'm trying to give you with this slide.
It's just a way to describe
sets that have special properties, and we'll see in the next couple of slides,
we're going to define a new commutative ring
and that's going to be the integers modulo m,
and it's kind of what you think based on what we've done this week
but we'll talk about that in a minute, okay?
So here's the definition of a commutative ring and fields.
We’ll also see fields later on. We're going to talk about the
set called the complex numbers, and we'll see that the complex numbers form a field.
But again, I'll save that for later.
Congruence classes, so now let's bring this back down to where we were, okay?
So rings, fields, we have these things in our back pocket when we need them.
What is a congruence class?
A congruence, or equivalence class modulo m of an integer a
is the set of integers defined as follows: so
we use square brackets and put a in the middle,
and we define this to be… so this “colon equals” means
“defined as” or 'defined to be' something along those lines.
[square bracket] a is defined to be the set of integers x such that
x is congruent to a mod m.
So for example, if m were 7 and my a were 1,
the set square bracket 1 would consist of the elements
1, 8, 15, dot dot dot,
and on the other side it would also consist of negative 6, negative 13, negative 20,
and all those as well. Okay?
So there's an example of the congruence class of 1 mod 7.
Further, we define this Z m notation.
So the integers modulo m, sometimes it's denoted by
a Z with a slash m Z.
In this course, we're going to use this first description, this Z
with the little subscript m.
Later on in your mathematical careers, you're more likely to see this notation,
for reasons which I won't get into now,
but it's a good notation so you should know it.
And this is defined to be what? Well this is defined to be
the set consisting of the sets
of 0, 1, all the way up to m minus 1,
and again these sets are taken modulo m.
You could argue that maybe we should decorate these square brackets with an m.
We're not going to do that, just to keep it simple.
Notice that you don't need to always use the symbols from
0, 1, all the way up to m minus 1. Remember that the equivalence class consisting of
0 is the same as the equivalence class consisting of m.
So if you wanted, you could write the set as 1, 2, 3, m minus 1, all the way up to m,
instead of - ignoring 0.
You can also use different elements as well. These elements are just the most
convenient ones to describe Z mod m. Okay?
So that's our congruence classes.
How do we put this all together? So I had this previous slide defining
rings and fields and I have this slide defining congruence classes.
Well the merger between the two is
given by giving the set Z m
a ring structure, okay,
and the ring structure is going to be what you think it is,
but I'm still going to define it formally.
So the ring Z m. So we're going to turn this set
into a ring by defining addition and subtraction
and multiplication as follows, okay?
So subtraction, again, you can think of subtraction as the additive inverse.
Now how do we do this? We take a plus or minus b,
and if I'm adding these - so remember, this is the set…
this is the congruence - this a is the congruence class of a,
and this b is the congruence class of b. These two things are sets.
So what we're trying to do is we're trying to define an operation on sets.
And on these two sets, how do we define the operation?
Well we take a plus b
to be the congruence class of a plus b, okay?
That doesn't look like we're doing very much, right? It looks like we're just pushing symbols,
but there's some going on here, right? We're adding two sets together.
How do we claim that we add the two sets together?
Take the arguments and add them.
And in the same way, we're going to multiply two sets together,
and how are we going to multiply these two sets together?
Well take the two arguments, a and b,
multiply them together, and look at that congruence class.
Something I'll get you to check is that this makes 0 the additive identity
and 1 the multiplicative identity.
And again, notice that a plus b means
add and then reduce modulo m. That's basically what's happening here.
You're adding first, and then reducing modulo m.
That's how the addition is defined on this set Z m.
And that's going to turn this into a ring. This is not something I'm going to
prove, but maybe this is something that you should…
well this is definitely something that you do.
Test, go back to the definition of a ring and try it out with this set,
and make sure you understand why this operation makes it a ring.
The members are sometimes called representative members,
so the members 0, 1, all the way up to m minus 1 are sometimes called representative members.
So that's like I said, right, we can change the 0 to m
and it doesn't really change the definition of the set,
it's still the same thing.
Something else to note, when m is a prime,
let’s call it p, the ring Z p is also a field
as non-zero elements have multiplicative inverses,
they're invertible.
We'll see this later.
We will see this later. I'll leave it at this for now,
but this is why I want to introduce the concept of a ring and a field
right now because it fits in quite nicely.