Algebra Answers

Let R be a left artinian ring and C be a subring in the center Z(R) of R. Show that Nil C = C ∩ rad R.

Let R be a finite-dimensional k-algebra, M be an R-module and E = EndRM. Show that if f ∈ E is such that f(M) ⊆ (rad R)M, then f ∈ rad E.

Let R be a finite-dimensional k-algebra which splits over k. Show that, for any field K ⊇ k, rad (RK) = (rad R)K.

Let M,N be finite-dimensional modules over a finite-dimensional k-algebra R. For any field K ⊇ k, show that MK and NK have a common composition factor as RK-modules iff M and N have a common composition factor as R-modules.

For any nonzero ring k and any group G, show that the group ring kG is von Neumann regular iff k is von Neumann regular, G is locally finite, and the order of any finite subgroup of G is a unit in k.

Show that statement "for any von Neumann regular ring k, any finitely generated submodule M of a projective k-module P is a direct summand of P" is equivalent to the fact that, if k is a von Neumann regular ring, then so is Mn(k) for any n ≥ 1.

system of equation:in a restaurant,some people have chosen the same menu. if each one pays 75000 ll the amount of 28000 ll is still needed. if each one pays 85000 ll the restaurant manager repays them 42000 ll find the number of guests and price of the menu per person?

For any von Neumann regular ring k, show that any finitely generated submodule M of a projective k-module P is a projective k-module.

For any von Neumann regular ring k, show that any finitely generated submodule M of a projective k-module P is a direct summand of P.

Show that, if group G is not {1} can be right ordered, then, for any domain k, A = kG is J-semisimple.