Direct Sum of 2 rings

In summary, the Direct Sum of 2 rings, denoted as R ⊕ S, is a new ring formed by combining the elements of two existing rings, R and S. It is different from the Cartesian Product of the two rings, as it has its own defined operations. The Direct Sum has several properties, including distributivity, commutativity, and associativity, and can be extended to any finite number of rings. Some applications of the Direct Sum include its use in abstract algebra, coding theory, and cryptography.
  • #1
kathrynag
598
0
Show that the direct sum of 2 nonzero rings is never an integral domain


I started by thinking about what a direct sum is
(a,b)(c,d)=(ac,bd)
(a,b)+(c,d)=(a+c,b+d)
We have an integral domain if ab=0 implies a=0 or b=0
 
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  • #2
Hint: The first property is pretty relevant. The second not so much.
 
  • #3
So we look at (a,b)(c,d) with (a,b) not zero and (c,d) not zero. Then multiplying together will never result in 0
 
  • #4
What about a=d=1, and b=c=0? Then you get (1,0)*(0,1) = (0,0).
 

1. What is the definition of the Direct Sum of 2 rings?

The Direct Sum of 2 rings, denoted as R ⊕ S, is a new ring formed by combining the elements of two existing rings, R and S. The elements of R ⊕ S are ordered pairs (r, s) where r ∈ R and s ∈ S, and the operations of addition and multiplication are defined component-wise.

2. How is the Direct Sum of 2 rings different from the Cartesian Product of the two rings?

While the Direct Sum of 2 rings and the Cartesian Product of the two rings may seem similar, they are distinct mathematical concepts. The Direct Sum is a ring with its own defined operations, while the Cartesian Product is simply a set of ordered pairs.

3. What are the properties of the Direct Sum of 2 rings?

The Direct Sum of 2 rings has several important properties, including distributivity, commutativity, and associativity. It is also a commutative ring if both R and S are commutative rings, and a zero-divisor free ring if both R and S are zero-divisor free.

4. Can the Direct Sum of 2 rings be extended to more than 2 rings?

Yes, the concept of the Direct Sum can be extended to any finite number of rings. For example, the Direct Sum of 3 rings, denoted as R ⊕ S ⊕ T, is formed by combining the elements of three rings R, S, and T in ordered triples (r, s, t) with defined operations.

5. What are some applications of the Direct Sum of 2 rings?

The Direct Sum of 2 rings has various applications in abstract algebra, including in the study of modules, vector spaces, and linear transformations. It can also be used to construct new rings with desired properties from existing ones. Additionally, the Direct Sum has applications in coding theory and cryptography.

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