Abstract Algebra Answers

Let фi : Gi ------> G1 x G2 x G3 x…….Gi x ……Gr be given by фi (gi)= (e1 ,e2 ,…..gi,…..er) where gi єGi and ej is the identity of Gj. Prove that this is a injective map.

Let F be an additive group of all continuous functions mapping IR into IR. Let IR be the additive group of real numbers, and let ф :F ------> IR be given by ф(f) = ∫_0^4▒〖f(x)dx〗 . Prove that f is a homomorphism?

Find the subring of the ring Zx Zthat is not an ideal of Zx Z

Let R be a commutative domain that is not a field. Show that not always R is not J-semisimple implies R is semilocal, if R is a noetherian domain.

Let R be a commutative domain and S^(−1)R be the localization of R at a multiplicative set S.
Is "rad S−1R ⊆ S−1(rad R)" ?

Let R be a commutative domain and S^(−1)R be the localization of R at a multiplicative set S.
Is "R ∩ rad S−1R ⊆ rad R" ?

Let R be a commutative domain and S^(−1)R be the localization of R at a multiplicative set S.
Is "rad R ⊆ R ∩ rad S−1R"?

Is true statement over arbitrary ring: "If R is a commutative ring or a left noetherian ring, then any finitely generated artinian left R-module M has finite length."

If R is a commutative ring or a left noetherian ring, show that any finitely generated artinian left R-module M has finite length.

Let J be a nilpotent right ideal in a ring R. If I is a subgroup of J such that I • I ⊆ I and J = I + J2, show that I = J.