Checking Ring Isomorphism: Z_9 and Z_3 + Z_3

In summary, the question is about determining if the rings Z_9 and Z_3+Z_3 are isomorphic. The conversation discusses using the fundamental theorem of finitely generated abelian groups and the concept of characteristics in determining isomorphism. The suggested approach is to look at the order of 1 in the underlying additive group, and it is determined that the characteristics of the two rings are not equal.
  • #1
kathrynag
598
0
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?
 
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  • #2
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.
 
  • #3
Well, this problem is in the section on rings, so would I use that?
 
  • #4
The rings have an underlying abelian group. Thus if the rings are isomorphic, so are the groups...
 
  • #5
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.
 
  • #6
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)?
 
  • #7
Their characteristics would be the same?
 
  • #8
kathrynag said:
Their characteristics would be the same?

Yes, so what are the characteristics of your two rings? Are they the same?
 
  • #9
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?
 
  • #10
kathrynag said:
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?
 

Related to Checking Ring Isomorphism: Z_9 and Z_3 + Z_3

1. Can you explain what a checking ring isomorphism is?

A checking ring isomorphism is a mathematical concept that compares two mathematical structures, in this case the groups Z_9 and Z_3 + Z_3, to determine if they are isomorphic or if they have the same structure and properties. It involves examining the elements, operations, and relationships within each group to see if they can be mapped onto each other in a one-to-one manner.

2. How do you check if Z_9 and Z_3 + Z_3 are isomorphic?

To check if Z_9 and Z_3 + Z_3 are isomorphic, you can first create a Cayley table for each group, which shows all possible combinations of elements and their corresponding operations. Then, you can compare the two tables and see if there is a one-to-one mapping between the elements. If there is, then the groups are isomorphic.

3. What are the basic properties of Z_9 and Z_3 + Z_3?

Z_9 and Z_3 + Z_3 are both groups, which means they have a defined set of elements and operations (in this case, addition and multiplication) that follow certain rules. Both groups have a neutral element (0), inverses for each element, and the associative and commutative properties. However, Z_9 has 9 elements while Z_3 + Z_3 has 6 elements.

4. Can you give an example of a one-to-one mapping between Z_9 and Z_3 + Z_3?

Yes, an example of a one-to-one mapping between Z_9 and Z_3 + Z_3 is: 1 in Z_9 maps to (1,0) in Z_3 + Z_3, 2 in Z_9 maps to (2,0) in Z_3 + Z_3, 3 in Z_9 maps to (0,1) in Z_3 + Z_3, 4 in Z_9 maps to (1,1) in Z_3 + Z_3, 5 in Z_9 maps to (2,1) in Z_3 + Z_3, 6 in Z_9 maps to (0,2) in Z_3 + Z_3, 7 in Z_9 maps to (1,2) in Z_3 + Z_3, 8 in Z_9 maps to (2,2) in Z_3 + Z_3, and 0 in Z_9 maps to (0,0) in Z_3 + Z_3.

5. What are some real-world applications of checking ring isomorphism?

One real-world application of checking ring isomorphism is in cryptography, where it is used to ensure that different encryption schemes have the same level of security. It is also used in coding theory to identify equivalent error-correcting codes. In chemistry, isomorphism is used to compare the structures of molecules and in computer science, it is used in data compression algorithms.

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