Abstract Algebra: Rings, Unit Elements, Fields

[S_Manifesto]

1) Show that (R,*,+) is a ring, where (x*y)=x+y+2 and (x+y)=2xy+4x+4y+6. Find the set of unit elements for the second operation.

I understand that the Ring Axioms is 1. (R,+) is an albein group. 2. Multiplication is associative and 3. Multiplication distributes. I just don't understand how to go about this. A First Course in Abstract Algebra by John Fraleigh fails to show any examples of this type.

2) Let f: Z[√d]→M be an application such that f)x+y√d=A where
[x y]
A = Matrix[yd x]

Show that f is an isomorphism of rings.

I understand that I have to check the conditions of it being isomorphic, but once again the book does not give examples of how to do so. It's hard to attempt problems when I don't know where to begin.

 

[haruspex]

It will help avoid confusion if you adopt different symbols for different operations instead of using + for the ring operation and for normal addition.
Let's start with showing * is associative, i.e. (a*b)*c = a*(b*c). Write that out using the definition of * you are given.

 

[S_Manifesto]

It will help avoid confusion if you adopt different symbols for different operations instead of using + for the ring operation and for normal addition.
Let's start with showing * is associative, i.e. (a*b)*c = a*(b*c). Write that out using the definition of * you are given.

What I came up with was (x*y)=(y*x) → x+y+2 = y+x+2 to show it's albein.
x+(y+z)=(x+y)+z → x+(2yz+4y+4z+6)=(2xy+4x+4y+6)+z and got 4xyz+8xy+8xz+8yz+16x+16y+16z+30 = 4xyz+8xy+8xz+8yz+16x+16y+16z+30

For x+(y*z)=(x+y)*(x+z) → x+(y+z+2)=(2xy+4x+4y+6)*(2xz+4x+4z+6) and got 2xy+2xz+8x+4y+4z+14 = 2xy+2xz+8x+4y+4z+14

 

[haruspex]

Looks fine (though I still wish you'd adopted a different symbol for one of the pluses).

 

[S_Manifesto]

Looks fine (though I still wish you'd adopted a different symbol for one of the pluses).

The question has a plus with a circle around it, I don't know how to make that symbol.

But that's correct?

 

[haruspex]

Yes, your solution looks fine.
If you click 'go advanced' then the capital sigma symbol from the options that appear you get a LaTex reference panel. In there, open Binary Operators. But for the purpose of the thread, you could have just chosen some character like %, provided you explained it.

 

[SammyS]

The question has a plus with a circle around it, I don't know how to make that symbol.

But that's correct?

Here ...

⊕

 

[Dick]

Show that (R,*,\oplus) is a ring, where x*y=x+y+2 and x\oplusy=2xy+4x+4y+6. Find the set of unit elements of the second operation.

I've showed that it is a ring, showing it is Albein, Associative, and Distributive, but I do not know how to find the inverse to find the unit elements.

All I know is that I need to replace y with x'

You meant the first operation *, right? If you've shown it's an abelian group then you already know there are inverses under \oplus. You should probably find the identity under * in your ring first. What is it?