Download Ring Theory Exam Past Paper

Ring Theory is a fundamental area of abstract algebra that studies algebraic structures known as rings and their properties. It plays a central role in pure mathematics and has applications in number theory, algebraic geometry, coding theory, and cryptography. This Ring Theory exam past paper is designed to help students revise core concepts, practice problem-solving techniques, and become familiar with common examination questions.

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Ring Theory Exam Past Paper

Above is the past paper download link

(Source: Meru University)

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Section A: Short Answer Questions

1. Define a ring.
   A ring is a non-empty set $R$ equipped with two binary operations, addition and multiplication, such that $(R,+)$ is an abelian group, multiplication is associative, and multiplication distributes over addition from both sides.

2. What is a commutative ring?
   A commutative ring is a ring in which the multiplication operation is commutative, meaning $ab=ba$ for all $a,b\in R$.

3. Define a unity (identity) in a ring.
   A unity in a ring is an element $1\in R$ such that $a\cdot 1=1\cdot a=a$ for all $a\in R$.

4. What is a zero divisor?
   A zero divisor is a non-zero element $a\in R$ such that there exists a non-zero element $b\in R$ with $ab=0$ or $ba=0$.

5. Give one example of a ring.
   The set of integers $\mathbb{Z}$ under usual addition and multiplication is a ring with unity and no zero divisors.

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Section B: Structured Questions

6. Explain the concept of a subring and state the subring test (10 marks).
   A subring is a subset $S$ of a ring $R$ that is itself a ring under the operations of $R$. For $S$ to be a subring, it must be non-empty and closed under subtraction and multiplication. The subring test states that if for all $a,b\in S$, both $a-b\in S$ and $ab\in S$, then $S$ is a subring of $R$.

7. Describe ideals and their importance in ring theory (10 marks).
   An ideal is a special subset $I$ of a ring $R$ that absorbs multiplication by elements of $R$. That is, for all $r\in R$ and $a\in I$, both $ra\in I$ and $ar\in I$. Ideals are crucial because they allow the construction of quotient rings and help classify ring structures.

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Section C: Essay Questions

8. Discuss homomorphisms between rings and their properties (20 marks).
   A ring homomorphism is a function $f:R\to S$ between two rings that preserves addition and multiplication. That is, for all $a,b\in R$, $f(a+b)=f(a)+f(b)$ and $f(ab)=f(a)f(b)$. If the rings have unity, a homomorphism may also preserve the identity element. The kernel of a ring homomorphism, defined as $\ker (f)=\{a\in R\mid f(a)=0\}$, is an ideal of $R$, while the image of $f$ is a subring of $S$. Ring homomorphisms help relate different rings and play a vital role in understanding their structural similarities.

9. Explain quotient rings and give an example (20 marks).
   Given a ring $R$ and an ideal $I\subseteq R$, the quotient ring $R/I$ is formed by the set of cosets $a+I$, where $a\in R$. Addition and multiplication are defined naturally on these cosets. Quotient rings simplify ring structures and are widely used in algebra. For example, the ring $\mathbb{Z}/n\mathbb{Z}$ is a quotient ring formed from the integers modulo $n$, where $n$ is a positive integer.

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Section D: Problem-Solving Question

10. Prove that every field is an integral domain (10 marks).
    A field is a commutative ring with unity in which every non-zero element has a multiplicative inverse. Suppose $ab=0$ in a field. If a≠0a \neq 0a=0, then multiplying both sides by a−1a^{-1}a−1 gives $b=0$. Hence, fields have no zero divisors and are therefore integral domains.
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