Answer to Question #183506 in Abstract Algebra for Anand

Question

Which of the following statements are true, and which are false? Give reasons for your 

answers.

 i) If k is a field, then so is k × k.

 ii) If R is an integral domain and I is an ideal of R, then Char (R) = Char (R / I)

 iii) In a domain, every prime ideal is a maximal ideal. 

 iv) If R is a ring with zero divisors, and S is a subring of R, then S has zero divisors. 

v) If R is a ring and f(x)∈R[x] is of degree n ∈N, then f(x) has exactly n roots in R.

Answer 1

Solution:

(i)True.

k×k is a field iff K satisfies that for all a,b∈K that a²+b²=0, then a=b=0.

(ii) False.

Char (R) must be a Prime rather it should not be equal to Char(R/I)

(iii) True.

A finite integral domain is a field.

Hence, if R/A is finite, then R/A is a field ⟹A is a maximal ideal.

(iv) False.

the statement only holds true if subring is a prime ideal.

(v) True.

Let R be a commutative ring where each degree n polynomial has at most n roots (n>0).

Suppose R is not a domain; then there are a,b∈R, both nonzero, with ab=0.

Consider now f(x)=(x−a)(x−b)=x²−(a+b)x+ab=x²−(a+b)x

This degree two polynomial has at least three roots, provided a≠b. Contradiction.