Ideal and Factor Ring: Proving AxB is an Ideal of RxS

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In summary, we need to show that A x B is an ideal of R x S and that every ideal C of R x S has the form C = A x B as shown in (a). To do this, we use the fact that A and B are ideals of R and S, respectively, and show that A x B is a ring. For part (b), we also use the fact that C is an ideal and R and S have identities, and show that C is a subset of A x B.
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hsong9
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Homework Statement



a) If A is an ideal of R and B is an ideal of S.
Show that A x B is an ideal of R x S.

b) Show that every ideal C of RxS has the form C = AxB as in(a)
[hint: A = { a in R | (a,0) in C}]



The Attempt at a Solution


a)Since A and B are ideal of R and S, aR and Ra are subsets of A, bS and Sb are subsets of B.
Let (a,b) in AxB and (r,s) in RxS, (a,b)(r,s) = (ar,bs) in AxB since ar in A and bs in B.

b) Let A = { a in R | (a,0) in C} and
B = { b in S | (0,b) in C}
We need to show that AxB = <(a,0),(0,b)>.
my idea is correct?
 
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  • #2
(a) looks good if you already have a theorem that A x B is a ring. If you don't have that theorem, then don't forget to do that routine step.

For (b), you have to show [tex]A\times B\subset C[/tex] and [tex]C\subset A\times B[/tex]

I regard [tex]A\times B=\langle(a,0),(0,b)\rangle\subset C[/tex] as the easier direction because you only need to use the fact that C is a ring. Really there's nothing to show.

To prove [tex]C\subset A\times B[/tex], I think you need to use the facts that C is an ideal and furthermore that R and S have identities. (Are you allowed to assume R and S have identities? I hope so.)
 

1. What is an ideal ring?

An ideal ring is a mathematical structure that consists of a set of elements and two operations, addition and multiplication, that follow certain rules. It is a generalization of the concept of a ring, which is a mathematical structure that has a set of elements and two operations, addition and multiplication, that also follow certain rules. The main difference between a ring and an ideal ring is that in an ideal ring, the elements can be multiplied by elements from outside the ring.

2. What is a factor ring?

A factor ring, also known as a quotient ring, is a mathematical structure that is obtained by dividing an ideal ring by one of its ideals. It consists of a set of cosets, which are subsets of the original ring that contain elements that are related by the ideal. The operations of addition and multiplication in the factor ring are defined based on the operations in the original ring, but they are restricted to the cosets.

3. What is the difference between an ideal and a factor ring?

An ideal is a subset of a ring that satisfies certain properties, such as closure under addition and multiplication. It is used to extend the operations of the ring to elements outside of the ring. A factor ring, on the other hand, is obtained by dividing an ideal ring by one of its ideals. It is a new mathematical structure that retains some of the properties of the original ring, but with restricted operations.

4. How are ideal and factor rings used in mathematics?

Ideal and factor rings are used in many areas of mathematics, such as abstract algebra, number theory, and algebraic geometry. They are useful for studying the structure and properties of rings, and they have applications in fields such as coding theory and cryptography. Ideal and factor rings also have connections to other mathematical structures, such as fields, modules, and algebras.

5. Can you give an example of an ideal and factor ring?

One example of an ideal ring is the ring of integers, and an example of an ideal in this ring is the set of all even integers. The factor ring obtained by dividing the ring of integers by this ideal is the ring of integers modulo 2, which consists of two cosets: the even integers and the odd integers. The operations of addition and multiplication in this factor ring are restricted to these two cosets, and the resulting structure is a field.

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