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**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. $\displaystyle \forall a \in I$ and $\displaystyle \forall r \in R$, $\displaystyle a+r \in I$

II. $\displaystyle \forall a \in I$ and $\displaystyle \forall r \in I$, $\displaystyle a*r \in I$

III. $\displaystyle \forall a \in R$ and $\displaystyle \forall r \in R$, $\displaystyle a+r \in I$

IV. $\displaystyle \forall a \in I$ and $\displaystyle \forall r \in R$, $\displaystyle a*r \in I$

V. $\displaystyle \forall a \in I$ and $\displaystyle \forall r \in I$, $\displaystyle a+r \in I$

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.