Ring with Unity: Subrings Isomorphic to Z & Z_m

In summary, the conversation discusses the possibility of a ring with unity containing two subrings that are isomorphic to Z_n and Z_m, respectively. It is determined that this is possible, with the example of Z_4 x Z_2 being given. It is also mentioned that the rings Zn and Z/Zn are isomorphic.
  • #1
ehrenfest
2,020
1
[SOLVED] rings with unity

Homework Statement


Corollary 27.18 (in Farleigh) tells us that every ring with unity contains a subring isomorphic to either Z or some Z_n. Is it possible that a ring with unity may simultaneously contain two subrings isomorphic to Z_n and Z_n with n not equal to m? If it is possible, give an example. If it is impossible, prove it.

EDIT: change the second Z_n to Z_m

Homework Equations


The Attempt at a Solution


My intuition tells me it is impossible. But I have no idea how to prove it.
 
Last edited:
Physics news on Phys.org
  • #2
What about Z/8Z? All of its subrings are isomorphic to some Zn. Does that mean all its subrings are isomorphic to each other?
 
  • #3
ehrenfest: Given that the meanings you give to the words is not the most common, you really need to be specific about their meaning.

e.g. I have absolutely no idea if, in this particular context, your use of 'subring' in this context requires the subring to contain a unit, and if that unit has to be the same as the one in the enclosing ring.
 
  • #4
A ring <R,+,*> is a set R together with two binary operations + and * such that the following axioms are satisfied:
1) <R,+> is an abelian group
2) Multiplication is associative
3) For all a,b,c in R, a*(b+c)=(b+c)*a=a*b+a*c

A subring is a subset of a ring that is also a ring.

See the EDIT.

Yes. I see. Z_8, we have the subring {0,4} which isomrphic to Z_2 and the whole ring is also a subring.
 
Last edited:
  • #5
ehrenfest said:
A ring <R,+,*> is a set R together with two binary operations + and * such that the following axioms are satisfied:
1) <R,+> is an abelian group
2) Multiplication is associative
3) For all a,b,c in R, a*(b+c)=(b+c)*a=a*b+a*c

A subring is a subset of a ring that is also a ring.

See the EDIT.

Yes. I see. Z_8, we have the subring {0,4} which isomrphic to Z_2 and the whole ring is also a subring.
The subalgebra {0, 4} of Z_8 is not isomorphic to Z_2.
 
Last edited:
  • #6
Why?
4+4=0
0+4=4
0+0=0
4*4=0
4*0=0
0*0=0

This is the same algebra as Z_2.

EDIT: you're right 1*1=1 not 0
EDIT: then what subring of Z_8 is isomorphic to Z_n where n is not equal to 8?
 
Last edited:
  • #7
Oops, that example doesn't work. Try Z_4 X Z_2.
 
  • #8
Then we have subrings isomorphic to Z_2 and Z_4. Thanks.
 
  • #9
is the ring Zn the same as the ring Z/Zn? (the one whose elements are modulo equivalence classes, with regular summation and multiplication?)
 
  • #10
jacobrhcp said:
is the ring Zn the same as the ring Z/Zn? (the one whose elements are modulo equivalence classes, with regular summation and multiplication?)

Yes, they are isomorphic since the kernel of

phi: Z -> Z_n defined by phi(z) = z mod n

is Zn.
 
  • #11
no I meant if Zn was defined in that way >_>

but it's not ... but almost. Thanks anyway. =)
 
  • #12
jacobrhcp said:
but it's not ... but almost.

What do you mean?
 

Related to Ring with Unity: Subrings Isomorphic to Z & Z_m

1. What is a subring?

A subring is a subset of a larger ring that still satisfies the properties of a ring. This means that the subring must be closed under addition and multiplication, and also contain the identity element.

2. What does it mean for a subring to be isomorphic?

Two rings are isomorphic if there exists a bijective homomorphism between them. In simpler terms, this means that the two rings have the same underlying structure and operations, but may have different elements.

3. What is the significance of Z & Z_m in relation to subrings isomorphic to Z & Z_m?

Z is the set of integers, while Z_m is the set of integers modulo m. These two sets are commonly used in subring isomorphism because they have simple and well-understood structures. Additionally, many other rings can be constructed from these sets.

4. Can a ring have multiple subrings isomorphic to Z & Z_m?

Yes, a ring can have multiple subrings that are isomorphic to Z & Z_m. This is because there can be multiple ways to construct a subring that satisfies the properties of a ring.

5. How is the concept of "ring with unity" related to subrings isomorphic to Z & Z_m?

A ring with unity is a ring that contains an identity element. This element is usually denoted as 1 and has the property that for any element a in the ring, 1*a = a*1 = a. Subrings isomorphic to Z & Z_m are often studied in relation to rings with unity because they can provide insight into the structure and properties of these rings.

Similar threads

  • Topology and Analysis
Replies
11
Views
2K
  • Calculus and Beyond Homework Help
Replies
6
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
8
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
3K
  • Calculus and Beyond Homework Help
Replies
1
Views
3K
  • Calculus and Beyond Homework Help
Replies
1
Views
3K
  • Calculus and Beyond Homework Help
Replies
8
Views
4K
  • Calculus and Beyond Homework Help
Replies
3
Views
2K
  • Calculus and Beyond Homework Help
Replies
12
Views
2K
Back
Top