• Support PF! Buy your school textbooks, materials and every day products Here!

Abstract Algebra: Rings, Unit Elements, Fields

  • #1
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.
 

Answers and Replies

  • #2
haruspex
Science Advisor
Homework Helper
Insights Author
Gold Member
33,533
5,451
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.
 
  • #3
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
 
  • #4
haruspex
Science Advisor
Homework Helper
Insights Author
Gold Member
33,533
5,451
Looks fine (though I still wish you'd adopted a different symbol for one of the pluses).
 
  • #5
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?
 
  • #6
haruspex
Science Advisor
Homework Helper
Insights Author
Gold Member
33,533
5,451
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.
 
  • #7
SammyS
Staff Emeritus
Science Advisor
Homework Helper
Gold Member
11,315
1,006
The question has a plus with a circle around it, I don't know how to make that symbol.

But that's correct?
Here ...

 
  • #8
Dick
Science Advisor
Homework Helper
26,258
618


Show that (R,*,[itex]\oplus[/itex]) is a ring, where x*y=x+y+2 and x[itex]\oplus[/itex]y=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 [itex]\oplus[/itex]. You should probably find the identity under * in your ring first. What is it?
 

Related Threads on Abstract Algebra: Rings, Unit Elements, Fields

  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
3
Views
3K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
3
Views
5K
  • Last Post
Replies
3
Views
2K
Replies
2
Views
892
Replies
2
Views
3K
Replies
3
Views
1K
Replies
1
Views
1K
Top