Checking Ring Isomorphism: Z_9 and Z_3 + Z_3

[kathrynag]

I was asked to decide if Z_9 and the direct sum of Z_3 and Z_3 are isomorphic.

Do I check to see if they are 1-1 and onto?

 

[snipez90]

Do you know the fundamental theorem of finitely generated abelian groups? This seems like a pretty big theorem but some abstract algebra texts make use of it early on without proof.

 

[kathrynag]

Well, this problem is in the section on rings, so would I use that?

 

[micromass]

The rings have an underlying abelian group. Thus if the rings are isomorphic, so are the groups...

 

[kathrynag]

I guess I have an easier time showing something is isomorphic when I have defined a mapping.
Let h:Z_9--->Z_3+Z_3 be defined by h(9n)=(3m,3n)*(3a,3b)
I guess if I have some kind of mapping I can justify my answer better.

 

[Rudeboy37]

I would suggest looking at the characteristic of each ring (i.e., the order of 1 in the underlying additive group)

If rings (or groups) are isomorphic, what must be true about their characteristics (or orders of elements)?

 

[kathrynag]

Their characteristics would be the same?

 

[Rudeboy37]

Their characteristics would be the same?

Yes, so what are the characteristics of your two rings? Are they the same?

 

[kathrynag]

Well a characteristic is the smallest positive integer n such that n*1=0
So for Z_9, we have 0*1=0, so 0 is the characteristic?
For the direct sum, we have (0,0)(1,1)=(0,0), so (0,0) is the characteristic?

 

[Rudeboy37]

Well a characteristic is the smallest positive integer n such that n*1=0

This is correct, but 0 nor 0,0 are positive integers. Remember that n*1 means 1+1+...+1 n-times. Basically, you can think of n*1 as 1^n in the underlying additive group. The characteristic of a ring can also be thought of as the order of 1 (the multiplicative identity element) in the underlying additive group (except if no positive integer n exists we say the characteristic is 0 where as we say the order is infinity).

So knowing this, what is the characteristic of your rings? Are they equal?