Page 201 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes 2. An ideal P of a ring R with identity is a prime ideal of R. If and only if the .................. R/P
is an integral domain.

(a) polynomial ring (b) subring
(c) quotient ring (d) ideal ring
3. If x  R, it has multiplicative inverse iff RX = ..................
(a) R (b) RX -1

(c) XR (d) X
4. A proper ideal m of a ring R is called maximal ideal of whenever I is an ideal of R such that
m .................. I .................. R then either I = m or I = R.
(a) ,  (b) , 

(c) ,  (d) , 
5. If R be a ring with identity. An ideal M in R is maximal if and only if .................. is a field.
(a) R.M (b) R/M
(c) M/R (d) R+M

19.3 Summary

The characteristic of any domain or field is either zero or a prime number.

The definition and examples of prime and maximal ideals.

The proof and use of the fact that a proper ideal I of a ring R with identity is prime

(or maximal) iff R/I is an integral domain (or a field).
Every maximal ideal is a prime ideal.

An element p of an integral domain R is prime iff the principal ideal pR is a prime ideal of

R.
Z, is a field iff n is a prime number.

The construction of the field of quotients of an integral domain.

19.4 Keywords

Prime Ideal: A ideal P of a ring R is called a prime ideal of R if whenever ab  P for a, b  R, then
either a  P or b  P.

Proper Ideal: A proper ideal M of a ring R is called a maximal ideal if whenever I is an ideal of
R such that M  I  R, then either I = M or I = R.
Maximal Ideal: Every maximal ideal of a ring with identity is a prime ideal.

19.5 Review Questions

1. Let F be a field. Show that F, with the Euclidean valuation d defined by d(a) = 1  a 
F/{0}, is a Euclidean domain.
2. Let F be a field. Define the function
d : F(x)\{0}  N  {0} : d(f(x)) = deg f(x).

Show that d is a Euclidean valuation on F[x], and hence, F[x] is a Euclidean domain.

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