1. ## Abstract Algebra Problem

I'm working on a practice exam for my major field exam, and I have not taken abstract algebra yet, so these problems are giving me trouble. I only need help with one of them though:

Let R be a ring with the binary operations + and * and let I be an ideal of R. Which of the following is/are true?

I. $\forall a \in I$ and $\forall r \in R$, $a+r \in I$
II. $\forall a \in I$ and $\forall r \in I$, $a*r \in I$
III. $\forall a \in R$ and $\forall r \in R$, $a+r \in I$
IV. $\forall a \in I$ and $\forall r \in R$, $a*r \in I$
V. $\forall a \in I$ and $\forall r \in I$, $a+r \in I$

I know that II and V are true, and I know that III is false, but I'm not entirely certain about IV (though I suspect it is true) and V.

2. ## Re: Abstract Algebra Problem

Originally Posted by Aryth
I'm working on a practice exam for my major field exam, and I have not taken abstract algebra yet, so these problems are giving me trouble. I only need help with one of them though:

Let R be a ring with the binary operations + and * and let I be an ideal of R. Which of the following is/are true?

I. $\forall a \in I$ and $\forall r \in R$, $a+r \in I$
II. $\forall a \in I$ and $\forall r \in I$, $a*r \in I$
III. $\forall a \in R$ and $\forall r \in R$, $a+r \in I$
IV. $\forall a \in I$ and $\forall r \in R$, $a*r \in I$
V. $\forall a \in I$ and $\forall r \in I$, $a+r \in I$

I know that II and V are true, and I know that III is false, but I'm not entirely certain about IV (though I suspect it is true) and V.
Statement one is false. Consider the ring of integers, and the ideal the set of even integers.

If you add and odd integer to and even integer you get an odd integer, so it is not back in the ideal.

IV is absolutely true. That is one of the defining characteristics of an ideal.

V. Ideals are subgroups under the addition operation so that must be true as well.

3. ## Re: Abstract Algebra Problem

Thanks. That makes sense. Glad you mentioned I, that last V was a mistake. Haha

4. ## Re: Abstract Algebra Problem

Originally Posted by TheEmptySet
Statement one is false. Consider the ring of integers, and the ideal the set of even integers.

If you add and odd integer to and even integer you get an odd integer, so it is not back in the ideal.

IV is absolutely true. That is one of the defining characteristics of an ideal.

V. Ideals are subgroups under the addition operation so that must be true as well.
Number II is true as well, since a subring is closed under multiplication. (didn't see the II from before)

After all, a condition for being an ideal is

1) nonempty subset
2) subring
3) satisfies the ideal conditions