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Let $R = R_1⊕R_2$ and let $I_1$ and $I_2$ be ideals of rings $R_1$ and $R_2$ respectively. We consider $R_1$ and $R_2$ as subrings of R under canonical embeddings $R_i \rightarrow R$ and thus $I_1$ and $I_2$ are considered as subrings of R.

(a) Prove that $I_1$ and $I_2$ are ideals of $R$.

(b) If each $I_i$ is a maximal ideal of $R_i$, is the ideal $I_1 +I_2$ of $R$ maximal? Prime?

I honestly don't know how to approach either. I think for a I want to show it just by using the definition, but that seems kind of tedious. Is there a more efficient way?

For b, I know that an ideal I is maximal if and only if R/I is a field and an ideal P is prime if and only if R/P is an integral domain

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  • $\begingroup$ Is the direct sum of fields a field? $\endgroup$ Dec 17, 2017 at 2:44
  • $\begingroup$ @AlfredYerger yes. how does that apply here? $\endgroup$
    – John Smith
    Dec 17, 2017 at 2:48
  • $\begingroup$ $R = R_1 \times R_2 = \{ (a,b), a \in R_1, b \in R_2\}$ is a ring with pointwise addition and multiplication, and it has many zero divisors. The embedding is $\iota : a \in R_1 \mapsto (a,0) \in R_1 \times R_2$. Simply check $\iota(I_1)$ satisfies the axioms of an ideal. $\endgroup$
    – reuns
    Dec 17, 2017 at 3:18
  • $\begingroup$ @reuns Why can you use $\times$ instead of direct sum? $\endgroup$
    – John Smith
    Dec 17, 2017 at 3:49
  • $\begingroup$ Because it is what $R_1 \oplus R_2$ means : $R_1 \times R_2$ with the pointwise addition and multiplication. $\endgroup$
    – reuns
    Dec 17, 2017 at 4:06

2 Answers 2

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The set $I_1\oplus 0$ is an ideal since if $x\oplus y$ is any element of $R$ and $a\oplus 0$ any element of the set, then $$(x\oplus y)(a\oplus 0)=xa\oplus 0\in I_1\oplus 0$$

However $I_1\oplus I_2$ need not be prime for example let $R_1=R_2=\mathbb{Z}$ and $$(2\oplus 1)(1\oplus 3)\in (2)\oplus (3)$$ but $2\oplus 1\not \in (2)\oplus 0$ and similarly.

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  • $\begingroup$ Thanks! Can you give some insight into how to do part a please? $\endgroup$
    – John Smith
    Dec 17, 2017 at 4:00
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    $\begingroup$ I answered that. $\endgroup$ Dec 17, 2017 at 4:01
  • $\begingroup$ Oh my bad, Is that all that needs to be done? $\endgroup$
    – John Smith
    Dec 17, 2017 at 4:06
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For part (b), the sum of two prime ideals is not necessarily prime. Consider $\mathbb{Z}$ with the ideals $\langle 2 \rangle, \langle 3 \rangle$.

As for maximality, is $I_1 + I_2$ an ideal? $I_1$ and $I_2$ are maximal, so what can you say about $I_1 + I_2$?

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  • $\begingroup$ is $I_1 + I_2$ an ideal? $\endgroup$
    – John Smith
    Dec 17, 2017 at 5:42
  • $\begingroup$ There doesn't seem to be anything remotely useful in either paragraph, not even hint-wise. $\endgroup$
    – rschwieb
    Dec 17, 2017 at 12:37
  • $\begingroup$ Sorry it seems as if I misread the problem. $\endgroup$
    – incertia
    Dec 17, 2017 at 15:20

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