Answer to Question #179498 in Abstract Algebra for swarnajeet kumar

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

i) If k is a field, then so is k × k. True. K is a subset of k*k .

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

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

iv) If R is a ring with zero divisors, and S is a subring of R, then S has zero divisors. False. This is not possible, because if x is a zero-divisor in S, then by definition x≠0 and there exists y∈S, y≠0, such that y=0, x=0. But then x and y are zero-divisors in RR as well.

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.

The proof is by induction on degree. If n=0, then f is a non-zero constant polynomial and therefore has no roots. Thus, the statement is true for n=0