Direct Sum vs Direct Product of Rings: Irish & Rosen

In summary, the conversation discusses the difference between direct sum and direct product of rings, with the Chinese Remainder Theorem being mentioned as an example. It is noted that direct sum and direct product coincide for finitely many summands, and the definition of both can be found on Wikipedia. The context is specifically focused on rings, not modules.
  • #1
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What's the difference (if any) between a direct sum and a direct product of rings?

For example, in Ireland and Rosen's number theory text, they mention that, in the context of rings, the Chinese Remainder Theorem implies that [tex]\mathbb{Z}/(m_1 \cdots m_n)\mathbb{Z}\cong\mathbb{Z}/m_1\mathbb{Z}\oplus\cdots\oplus\mathbb{Z}/m_n\mathbb{Z}[/tex]. The way I've been taught is that every [tex]\oplus[/tex] should be replaced with a [tex]\times[/tex] so that we are talking about direct products, not direct sums.
 
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  • #2
Yes, direct sum and direct product coincide for finitely many summands. You can check out the definition of both product on wikipedia.
 
  • #3
He's not dealing with modules -- he's dealing with rings.
 

FAQ: Direct Sum vs Direct Product of Rings: Irish & Rosen

What is the difference between direct sum and direct product of rings?

The direct sum and direct product of rings are both ways of combining two or more rings to create a new ring. The main difference is that in a direct sum, the elements of the new ring are pairs of elements from the original rings, while in a direct product, the elements are functions that map from the original rings to the new ring.

How are the operations of addition and multiplication defined in direct sum and direct product of rings?

In a direct sum, addition and multiplication are defined component-wise, meaning that for two elements (a,b) and (c,d) in the direct sum, their sum is (a+c, b+d) and their product is (ac, bd). In a direct product, addition and multiplication are defined point-wise, meaning that for two functions f and g, their sum is the function h(x) = f(x) + g(x) and their product is the function k(x) = f(x) * g(x).

What is the relationship between direct sum and direct product of rings?

The direct sum and direct product of rings are related in that the direct sum is a special case of the direct product. Specifically, a direct sum of two rings A and B can be seen as a direct product of A and B, where the elements of A and B are viewed as functions that map to a new ring C.

Can you give an example of a direct sum and direct product of rings?

One example of a direct sum is the direct sum of the rings Z and Z, denoted as Z⊕Z. The elements of this direct sum are pairs (a,b) where a and b are integers. Addition and multiplication are defined as (a,b) + (c,d) = (a+c, b+d) and (a,b) * (c,d) = (ac, bd). An example of a direct product is the ring Z×Z, where the elements are functions f and g that map from Z to Z. Addition and multiplication are defined as f + g = h, where h(x) = f(x) + g(x) and f * g = k, where k(x) = f(x) * g(x).

What are the applications of direct sum and direct product of rings in mathematics?

The direct sum and direct product of rings have various applications in abstract algebra, number theory, and algebraic geometry. They are used to construct new rings with specific properties, to study properties of existing rings, and to define and study algebraic structures such as modules and algebras. They also have applications in coding theory, cryptography, and other areas of mathematics and computer science.

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