Abstract Algebra Answers

Which of the following statements are true, and which are false? Give reasons for your answers. i) For any ring R and . a, b R, (a b) a 2ab b 2 2 2 ∈ + = + + ii) Every ring has at least two elements. iii) If R is a ring with identity and I is an ideal of R, then the identity of R I is the same as the identity of R. iv) If f : R → S is a ring homomorphism, then it is a group homomorphism from (R, +) to (S, +). v) If R is a ring, then any ring homomorphism from R ×R into R is surjective

a) Prove that every non-trivial subgroup of a cyclic group has finite index. Hence 

prove that (Q, +) is not cyclic. (7) 

 b) Let G be an infinite group such that for any non-trivial subgroup H of 

G, G : H < ∞. Then prove that 

 i) 

H ≤ G ⇒ H = {e} or H is infinite; 

 ii) If g ∈G, g ≠ e, then o(g) is infinite. (5) 

 c) Prove that a cyclic group with only one generator can have at most 2 elements. (3) 

Find all left cosets of the subgroup {ρ₀,ρ₂} = {e,(1,3)(2,4)} in D₄

Prove that for every subgroup H of S_n, either all of the permutations in H are even, or exactly half are even.

Construct a field with 125 elements.

Check whether or not Q[x] / < 8x³ + 6x² − 9x + 24 > is a field

Find all the units of Z[ √− 7].

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.

a group g is isomorphism to one of its proper subgroup then g=z is it true z=integers geoup

Suppose that R is a ring with identity such that char R=n>0 .if n is not prime, show that char R=n>0.if n is not prime, show that R has divisors of zero