Is there a ring without unity which can be expressed as a union of its three proper ideals? [closed]

Are there any rings without unity which can be expressed as a union of its three proper ideals?

closed as off-topic by user26857, Zev Chonoles, 6005, Watson, Parcly TaxelSep 16 '16 at 12:04

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If your rings are unital, then the answer is no, since $1\in R$ will then be contained in a proper ideal. If you do not insist on a unit, take $R=\mathbb Z_2\times \mathbb Z_2$ with zero multiplication. Then $I_1=\{(0,0),(1,0)\}$, $I_2=\{(0,0),(0,1)\}$, and $I_3=\{(0,0),(1,1)\}$ will work.
• By zero multiplication, do you mean that $1.1 = 1.0 = 0.1 = 0$? – Prince Kumar Sep 16 '16 at 7:32
• @PrinceKumar: zero multiplication means $x\cdot y = 0$ for any $x$ and $y$. – Keith Kearnes Sep 16 '16 at 17:58