Questions about Proving R/I is not the Zero Ring

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In summary, the conversation discusses the proof attached and raises questions about the notation and computation involved in working with ideals. It also clarifies that in the quotient ring R/I, the elements are of the form a + I.
  • #1
Artusartos
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I have a question about the proof that I attached...

1) Since R/I is not the zero ring, we know that [tex]1 \not= 0[/tex]. What is the reason to say [tex]1 + I \not= 0 + I[/tex] instead of [tex]1 \not= 0[/tex]?2) Also, how do we compute something like (a+I)(b+I)? Isn't this correct [tex](a+I)(b+I) = ab+aI+bI+I^2[/tex]?

3) Finally, if we have something like R/I, how do we know if the elements in R/I are of the form a+I or aI? Or is it both (since it's a ring)?

Thank you in advance
 

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Also, how do we compute something like (a+I)(b+I)? Isn't this correct [tex](a+I)(b+I) = ab+aI+bI+I^2[/tex]?

Strictly, yes; but you didn't finish. How can you simplify it further, using what you know about ideals and multiplication?

Finally, if we have something like R/I, how do we know if the elements in R/I are of the form a+I or aI? Or is it both (since it's a ring)?

R/I is the quotient of R by the normal subgroup I of (R,+) satisfying the additional constraint that RI = I. The cosets are of the form a + I.

What is the reason to say [tex]1 + I \not= 0 + I[/tex]

We know that [itex]0 \in I[/itex] by definition, so [itex]0 + I = I[/itex]. We also know that [itex]I[/itex] is a proper ideal, so it can't contain 1 (what happens if an ideal contains a unit?). It follows that [itex]1 + I \neq I[/itex].
 

Related to Questions about Proving R/I is not the Zero Ring

1. How do you prove that R/I is not the zero ring?

To prove that R/I is not the zero ring, we need to show that there exists an element in R/I that is not equal to zero. This can be done by finding an element in R that is not in the ideal I, and showing that its coset in R/I is not equal to the zero coset.

2. What is the significance of proving that R/I is not the zero ring?

Proving that R/I is not the zero ring is important because it helps us understand the structure and properties of the quotient ring. It also allows us to determine if the ideal I is a proper ideal or not.

3. Can R/I be the zero ring if R is not the zero ring?

No, if R is not the zero ring, then R/I cannot be the zero ring. This is because the zero coset in R/I is the image of the zero element in R, and since R is not the zero ring, there exists at least one element in R that is not equal to zero.

4. How does the existence of a unit element in R affect the proof that R/I is not the zero ring?

If R has a unit element, then proving that R/I is not the zero ring becomes easier. This is because we can choose the element in R that is not in the ideal I to be the unit element, and its coset in R/I will not be equal to the zero coset.

5. What are some common methods used to prove that R/I is not the zero ring?

Some common methods used to prove that R/I is not the zero ring include finding a non-zero element in R that is not in the ideal I, showing that its coset in R/I is not equal to the zero coset, and using the fact that R/I is a field if and only if I is a maximal ideal in R.

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