Abstract Algebra Answers

Let G be a finite group whose order is a unit in a ring k, and let W ⊆ V be left kG-modules. If V is projective as a k-module, then V is projective as a kG-module.

Let G be a finite group whose order is a unit in a ring k, and let W ⊆ V be left kG-modules. If W is a direct summand of V as k-modules, then W is a direct summand of V as kG-modules.

Let A be a normal elementary p-subgroup of a finite group G such that the index of the centralizer CG(A) is prime to p. Show that for any normal subgroup B of G lying in A, there exists another normal subgroup C of G lying in A such that A = B × C.

Let V be a kG-module and H be a subgroup in G of finite index n not divisible by char k. Show the following: If V is semisimple as a kH-module, then V is semisimple as a kG-module.

Let R be a subring of a right noetherian ring Q with a set S ⊆ R ∩ U(Q) such that every element q ∈ Q has the form rs^−1 for some r ∈ R and s ∈ S. Show that:the converse of " if Q is prime (resp. semiprime), then so is R " is true even without assuming Q to be right noetherian.

For R be a subring of a right noetherian ring Q=RS^-1 with a set S ⊆ R ∩ U(Q). Show that: if Q is semiprime, then so is R.

Let R be a subring of a right noetherian ring Q with a set S ⊆ R ∩ U(Q) such that every element q ∈ Q has the form rs^−1 for some r ∈ R and s ∈ S. Show that: if Q is prime, then so is R.

Let R be a subring of a right noetherian ring Q with a set S ⊆ R ∩ U(Q) such that every element q ∈ Q has the form rs^−1 for some r ∈ R and s ∈ S. Show that: if B is an ideal of R, then BQ is an ideal of Q.

Show that R =
Z nZ
Z Z
is not isomorphic to the prime ring P = M2(Z) if n > 1.

Show that R' =
Z nZ
0 Z
is not prime ring.