How to prove an ideal of a ring R which is defined as a coordinates

Therefore, $I$ cannot be a subring of $R$. In summary, the conversation discusses proving that $I$ is an ideal of $R$, but it is determined that $I$ is not a subring of $R$ due to issues with multiplication.
  • #1
cbarker1
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MHB
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Hi Everyone,

I am wondering how to prove an ideal of a ring $R$ which is defined as a coordinates. Let $R$ be the ring of $\mathbb{Z} \times \mathbb{Z}$. Let $I={(a,a)| a\in \mathbb{Z}}$. I determine that the $I$ is a subring of $R$. Next step is to show the multiplication between the elements of $R$ and $I$. But I have read in the book that I need worried about the elements of $R$ and not just $I$. Thanks,
Cbarker1
 
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  • #2
Hi Cbarker1,

Assuming that multiplication in $R$ is defined as $(a_{1},a_{2})\cdot (b_{1},b_{2}) = (a_{1}b_{1},a_{2}b_{2}),$ $I$ is not an ideal of $R$. For example $(1,0)\cdot(a,a) = (a,0)\notin I$ for any $a\neq 0$.
 

1. What is an ideal of a ring R?

An ideal of a ring R is a subset of the ring that satisfies certain properties. It is a set of elements that can be multiplied by any element in the ring and the result will still be in the ideal. Additionally, the ideal must be closed under addition and contain the additive identity element of the ring.

2. How is an ideal defined as coordinates?

An ideal of a ring R can be defined as coordinates by using the ring's operations of addition and multiplication to represent points on a coordinate plane. The elements of the ideal can be thought of as coordinates on the x-axis, while the elements of the ring can be thought of as coordinates on the y-axis.

3. What is the process for proving an ideal of a ring R?

The process for proving an ideal of a ring R involves showing that the ideal satisfies the properties of an ideal. This includes proving that the ideal is closed under addition and multiplication by elements of the ring, and that it contains the additive identity element of the ring. Additionally, it may be necessary to prove that the ideal is a subset of the ring.

4. Can an ideal of a ring R be proven using algebraic equations?

Yes, an ideal of a ring R can be proven using algebraic equations. This involves using the properties of the ideal and the ring's operations of addition and multiplication to manipulate equations and show that they hold for all elements in the ideal.

5. Are there any common mistakes to avoid when proving an ideal of a ring R?

Yes, there are some common mistakes to avoid when proving an ideal of a ring R. These include assuming that the ideal is a subset of the ring without proof, using incorrect operations or properties of the ideal, and not considering all elements of the ideal in the proof. It is important to carefully follow the definition of an ideal and use algebraic manipulations correctly in the proof.

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