Page 193 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes 2. Zp is an integral domain if p is a .................... number.
(a) even (b) odd

(c) prime (d) integer
3. A ring R has .................... zero divisor if and only if the cancellation law for multiplication
holds in R.

(a) 1 (b) 2
(c) 0 (d) 3
4. A ring (R, +,.) is called a .................... if (R | { 0 }.) is an abelian group.
(a) field (b) domain

(c) range (d) ideal
5. Every .................... integral domain is a field.
(a) infinite (b) finite
(c) direct (d) indirect

18.3 Summary

The definition and examples of an integral domain.

The definition and examples of a field.

Every field is a domain.

A finite domain is a field.

The characteristic of any domain or field is either zero or a prime number.

18.4 Keywords

Zero Divisor: A non-zero element a in a ring R is called a zero divisor in R if there exists: a non-
zero element b in R such that ab = 0.

Prime Number: Z is an integral domain iff p is a prime number.
p
Abelian Group: A ring (R, +,.) is called a field if (R\{0},.) is an abelian group.

18.5 Review Questions

1. Let n  N and m | n | < m < n. Then show that m is a zero divisor in Z .
n
2. List all the zero divisors in Z.
3. For which rings with unity will 1 be a zero divisor?
4. Let R be a ring and a  R be a zero divisor. Then show that every element of the principal
ideal Ra is a zero divisor.
5. In a domain, show that the only solutions of the equation x = x are x = 0 and x = 1.
2
6. Prove that 0 is the only nilpotent element in a domain.

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