Rings, Fields, and Ideals

Rings are mathematical objects which can be thought of as sets closed under the familiar operations of addition, subtraction, and multiplication. The technical definition involves the concept of a group, which I’ll take on maybe in a future post, so for now instead of defining rings rigorously I’ll instead rely on intuition and examples (also, in this post all rings considered will be commutative – note that this is not the most general case). The first, and perhaps most intuitive example of a ring are the ordinary integers (symbolized by \mathbb{Z}). When we add, subtract, or multiply two integers, we get another integer. Here are a few examples:

1+1=2

2+3=5

10-21=-11

-5\times3=-15

In each of these examples, whenever the two operands on the left hand side are integers, then the right hand side is also an integer.

But we also know of a fourth arithmetic operation, the operation of division, which we did not require rings to have. Surely, in a ring, some elements may divide others, for example, once again in the ring of integers:

10\div2=5

But it doesn’t hold for all elements (obviously it will not hold for any number being divided by zero, so when we speak of division, we mean only division by nonzero elements). For example,

10\div20=0.5

The right hand side, 0.5, is not an integer, despite 10 and 20 being both integers. This is what we mean when we say that a ring (here the ring of integers) need not be closed under the operation of division. When a ring is closed under the operation of division, we call it a field. The rational numbers \mathbb{Q} form a field. The sum, difference, product, and quotient of two rational numbers is always another rational number. The same also applies to the real numbers \mathbb{R} and the complex numbers \mathbb{C}, so they also form fields.

Another very important example of a ring is the ring of polynomials (of positive degree) in one variable x, familiar from high school mathematics:

0.01x

x^{2}-x+5

-7x^{5}-x^{4}+100x

For rings of polynomials in one variable x, we shall use the symbols \mathbb{Z}[x], \mathbb{Q}[x], \mathbb{R}[x] and \mathbb{C}[x] for polynomials with integer, rational, real, and complex coefficients respectively. All of these form rings – they are closed under addition, subtraction, and multiplication.

We now mention an important concept related to rings, called ideals of a ring. Ideals are subsets of a ring which are closed under addition and subtraction among its own elements, and multiplication by elements of the ring to which it belongs. We shall demonstrate with an example. Consider the set of integer multiples of 5, in the ring of integers \mathbb{Z}. We will symbolize this set by (5). Some of its elements include the following:

...-20,-15,-10, -5, 0, 5, 10, 15,20...

Now we show a few examples of what it means for (5) to be closed under addition and subtraction among its own elements and multiplication by elements of the ring \mathbb{Z} to which it belongs. First, we take two elements of (5), i.e. two multiples of 5, and add or subtract them together:

5+10=15

10-15=-5

100-65=35

We can see that whenever two multiples of 5 are added or subtracted from each other, the result is always another multiple of 5. Now we show examples of how (5) is closed under multiplication by any integer, i.e. any element of the ring \mathbb{Z}:

3\times10=30

-4\times15=-60

-7\times-5=35

Note that the integer (the first factor on the left hand side) which multiplies the multiple of 5 (the second factor on the left hand side) does not need to be a multiple of 5 itself; however their product (on the right hand side) is always a multiple of 5. This is reminiscent of the concept of vector spaces which are very useful not only in mathematics but also in physics and engineering – vector spaces are sets closed under addition among themselves and under multiplication by a “scalar”. In fact, both vector spaces and ideals of a ring are both special cases of more general mathematical objects called modules.

And for now that’s it. I would like to note that the way this post was written is not the way we do mathematics at higher levels. Mathematics is really built more on logic, rigor, and definitions, not just intuition and examples like what was done here (although intuition and examples can be useful). But I guess I wanted this post to be more accessible to people who don’t have formal mathematical training; I even used the division symbol “\div” which is almost never used in higher level mathematics. Ultimately it’s part of what I wanted to do in this blog – to write in a more accessible language and hopefully show more of what’s in higher level mathematics (and also physics) to the layperson. Since the other (and primary) purpose of this blog is to help me learn, many future posts might not be as accessible as this one, and hopefully feature more rigor; nevertheless I still want to continue writing expository posts such as these from time to time.

Also, I want to say that I’m still very much in the process of learning, and there might be mistakes in my expositions (aside from the confusing language). That’s why I always include references, which are written by people with more expertise (and more time to proofread). A reader of this blog may take the content of the posts as merely an invitation to read the references. In any case, reading the references I usually list at the end of my posts is always very much encouraged.

References:

Ring on Wikipedia

Field on Wikipedia

Ideal on Wikipedia

Module on Wikipedia

Algebra by Michael Artin

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9 thoughts on “Rings, Fields, and Ideals

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