# Prove these subrings are ideals

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

• Is the direct sum of fields a field? Dec 17, 2017 at 2:44
• @AlfredYerger yes. how does that apply here? Dec 17, 2017 at 2:48
• $R = R_1 \times R_2 = \{ (a,b), a \in R_1, b \in R_2\}$ 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. Dec 17, 2017 at 3:18
• @reuns Why can you use $\times$ instead of direct sum? Dec 17, 2017 at 3:49
• Because it is what $R_1 \oplus R_2$ means : $R_1 \times R_2$ with the pointwise addition and multiplication. Dec 17, 2017 at 4:06

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.

• Thanks! Can you give some insight into how to do part a please? Dec 17, 2017 at 4:00
• I answered that. Dec 17, 2017 at 4:01
• Oh my bad, Is that all that needs to be done? Dec 17, 2017 at 4:06

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 $I_1 + I_2$?

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