Ring Theory Questions – Daily Mathematics MCQs
Practise Ring Theory multiple-choice questions covering rings, ideals, homomorphisms, integral domains and fields. Attempt each problem before revealing its detailed mathematical solution.
About These Ring Theory Questions
Use this question bank for focused practice. Each MCQ is presented in crawlable HTML and includes answer choices, an instant answer check and a detailed solution that you can reveal after attempting the problem.
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Ring Theory Question Archive
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Question RT-001
Detailed Solution
The kernel is $\ker\varphi=\{r\in R:\varphi(r)=0_S\}$. If $a,b\in\ker\varphi$, then $\varphi(a-b)=\varphi(a)-\varphi(b)=0$, so $a-b\in\ker\varphi$. For any $r\in R$ and $a\in\ker\varphi$, $\varphi(ra)=\varphi(r)\varphi(a)=0$ and similarly $\varphi(ar)=0$. Hence $\ker\varphi$ is an ideal of $R$. Thus $\boxed{\mathrm{B}}$.Have a doubt or a different approach? Continue the mathematics discussion on Telegram.
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Question RT-002
Detailed Solution
For a positive integer $n$, $\mathbb Z_n$ is an integral domain exactly when $n$ is prime. Since $5$ is prime, $\mathbb Z_5$ is a field and hence an integral domain. The other choices have composite moduli and therefore contain zero divisors. Thus $\boxed{\mathrm{C}}$.Have a doubt or a different approach? Continue the mathematics discussion on Telegram.
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Prepare Smarter with Ring Theory MCQs
For competitive mathematics examinations, knowing a definition is only the first step. These questions are designed to make you apply theorems, identify subtle distinctions and practise the kind of conceptual reasoning that is useful in IIT JAM, CUET-PG, CSIR-NET, UGC-NET and GATE preparation.
For deeper study, combine this practice bank with the mathematics lecture videos, notes and other resources available on Fractal Frontier Maths.