Abstract Algebra Answers

If R is a commutative noetherian ring and P is a prime ideal of R, then prove that an R-module is P-primary if and only if each nonzero submodule of M is subisomorphic to R/P.

let R be comutetive ring with unity in which each ideal prime then show that R is a field.

 Showthatthe Polynomial x7-1oxs+15x+5 is not Solvable by ..

f is mapping from M to N . f is an R homomorphism and M is finitely generated then prove that N is also finitely generated.

Prove that Homz(Z,M) is isomorphic to M where M is an abelian group.

Prove that if N is a submodule of a finitely generated module M over a ring R .Then M/N is also finitely generated R module.

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