Abstract Algebra Answers

a frog lies at the bottom of a well whose depth is 30m. It starts to climb, rising up 2/5 every day and falling back 1/5 every night. How long will it take him to climb out of the well?

Does this hold over an arbitrary ring R: any direct product of copies of V is semisimple?

Let U, V be semisimple modules over a commutative ring. Is HomR(U, V ) also semisimple?

Let U, V be modules over a commutative ring R. If U or V is semisimple, show that U ⊗R V is also semisimple.

Show that for a semisimple module M over any ring R, the following conditions are equivalent:
(1) M is finitely generated;
(2) M is noetherian;
(3) M is artinian;
(4) M is a finite direct sum of simple modules.

Let R be a right semisimple ring. For x, y ∈ R, show that Rx =Ry iff x = uy for some unit u ∈ U(R).

Let R be a (left) semisimple ring. Show that, for any right ideal I and any left ideal J in R, IJ = I ∩ J. If I, J, K are ideals in R, prove the following distributive law: I ∩ (J + K) = (I ∩ J) + (I ∩ K)

Let R be the (commutative) ring of all real-valued continuous functions on [0, 1]. Is R a semisimple ring?