# Ring with Unity: Subrings Isomorphic to Z & Z_m

• ehrenfest
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.
ehrenfest
[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

## The Attempt at a Solution

My intuition tells me it is impossible. But I have no idea how to prove it.

Last edited:
What about Z/8Z? All of its subrings are isomorphic to some Zn. Does that mean all its subrings are isomorphic to each other?

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.

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:
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:
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:
Oops, that example doesn't work. Try Z_4 X Z_2.

Then we have subrings isomorphic to Z_2 and Z_4. Thanks.

is the ring Zn the same as the ring Z/Zn? (the one whose elements are modulo equivalence classes, with regular summation and multiplication?)

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.

no I meant if Zn was defined in that way >_>

but it's not ... but almost. Thanks anyway. =)

jacobrhcp said:
but it's not ... but almost.

What do you mean?

## 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.

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