Abstract Algebra Answers

Which of the following statements are true? Justify your answers. (This means that if you

think a statement is false, give a short proof or an example that shows it is false. If it is

true, give a short proof for saying so.)

(i)Given any ring R, there is an ideal I of R such that R/I is commutative.

(ii)If S is an ideal of a ring R and f a ring homomorphism from R to a ring R', then f-¹(f(S))=S.

Prove that if M ,N are Noetherian submodule of P then M +N is also Noetherian submodule of P.

M and N are Noetherian then their direct sum is also Noetherian

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.