Abstract Algebra Answers

Check if matrix A=

1 a b

0 1 c

0 0 1

Where a,b,c∈R

is an abelian group with respect to matrix multiplication.

Let R and S be rings and f : R→ S be a homomorphism. If x is an idempotent in R,

show that f(x) is an idempotent in S. Hence, or otherwise, determine all ring

homorphisms from Z×Z to Z .

Prove that every ideal I of a ring R is the kernel of a ring homomorphism of R.

Let F be the ring of all functions from R to R w.r.t. pointwise, addition and

multiplication. Let S be the set of all differentiable functions in F. Check whether S

is

i) a subring of F,

ii) an ideal of F.

Let R be a ring, I an ideal of R, J an ideal of I. Show that if J has a unity, then J is

an ideal of R. Also give an example to show that if J does not satisfy this condition

it need not be an ideal of R.

If U(R) denotes the group of units of a ring R, show that

U(R₁ × R₂) = U(R₁) × U(R₂) for rings R₁ and R₂.

Use FTH to determine all homomorphic images of D₈ upto isomorphism.

Define f:Z→Z_m × Z_n :f (x)=(xmodm,xmod n),m,n∈N.

i) If (m, n) = (3, 4), find Ker f.

ii) If (m, n) = (6, 4), find Ker f.

iii) What can you generalize about Ker f from (i) and (ii)?

Can there be a homomorphism from Z₈ ⊕Z₂ onto Z₄⊕Z₄ ? Give reasons for your

answer.

Prove, by contradiction, that A₄ has no subgroup of order 6.