Ring with Unity: Subrings Isomorphic to Z & Z_m

[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

Homework Equations

The Attempt at a Solution

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

 

[zhentil]

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

 

[Hurkyl]

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.

 

[ehrenfest]

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.

 

[Hurkyl]

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.

 

[ehrenfest]

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?

 

[zhentil]

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

 

[ehrenfest]

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

 

[jacobrhcp]

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

 

[ehrenfest]

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.

 

[jacobrhcp]

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

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

 

[ehrenfest]

but it's not ... but almost.

What do you mean?