Abstract Algebra Answers

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.

Let R be a commutative Q-algebra generated by x1, x2, . . . with the relations xixj = 0 for all i, j. Show that R is semiprimary (that is, rad R is nilpotent, and R/rad R is semisimple), but neither artinian nor noetherian.

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If R is a ring with identity, then prove that

<a> =\left \{ \sum_{finite}^{n} s_{ia}t_{i}; s_{i}, t_{i}\epsilon R\right \} =RaR

evatuate \int_{}^{c} \ f(z)dz, f(z)=\frac{z^{2}}{z+3} , c=\left | z \right |=1