How Do I Prove Ring Properties and Understand Their Structures?

[DanielThrice]

I'm working with elementary rings, and my professor gave me about ten of these to start but it seems like a lot of work with how he managed it. I know you guys don't answer homework so I chose so I can do the others. Any help would be greatly appreciated, the groups were easy but the rings are a little more difficult for me to grasp.

Consider the set R = {a + b (SQRT 2): a, b are in Z}
Prove that R is a subring of R.
Is R commutative?
Does R have identity?
Is R an integral domain?
Is R a division ring?
Is R a skew field?
Is R a field?

Can someone explain how I go about doing these processes?

 

[HallsofIvy]

I'm working with elementary rings, and my professor gave me about ten of these to start but it seems like a lot of work with how he managed it. I know you guys don't answer homework so I chose so I can do the others. Any help would be greatly appreciated, the groups were easy but the rings are a little more difficult for me to grasp.

Consider the set R = {a + b (SQRT 2): a, b are in Z}
Prove that R is a subring of R.

It's a really bad idea to use the same symbol, R, to mean two different things! I am going to use "R" to mean the usual real numbers and R' to mean this new set.

Is R commutative?
Does R have identity?
Is R an integral domain?
Is R a division ring?
Is R a skew field?
Is R a field?

Can someone explain how I go about doing these processes?

Yes- do the arithmetic! any thing in this new set, R', is of the form $a+ b\sqrt{2}$ where a and b are integers. What is the sum of two such things, $a+ b\sqrt{2}$ and $c+ d\sqrt{2}$? What is their product? Is $(a+ b\sqrt{2})(c+ d\sqrt{2})= (c+ d\sqrt{2})(a+ b\sqrt{2})$? If, for any a, b, it were true that $(a+ b\sqrt{2})(x+ y\sqrt{2})= a+ b\sqrt{2}$, what would x and y have to be?

As for the others, what are the definitions of "integral domain", "division ring", "skew field", and "field"?

 

[AlephZero]

You just have to apply the definitions, and see if they are true or false for the mathematical object that was defined in each question.

You are right, rings, integral domains, fields, etc are rather more complicated than groups.

The point of working through ALL the different examples yourself is so you begin to see what are the differences between the various mathematical structures, so you can relate the "abstract" ideas that will come up later in the course to these "concrete" examples, and invent more examples of your own to test out conjectures, etc.

Doing this may seem tedious, but if if you don't get a solid undertanding of what rings, integral domains, etc "really are" and what are the differences between them, it is more likely you will get lost later in the course because "you can't the forest becaose of all the trees".