Abstract Algebra Answers

Show that a ring R is semiprime iff, for any two ideals A,B in R, AB = 0 implies that A ∩ B = 0.

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
0 Z
is not prime ring.

For any integer n > 0, show that
R =
Z nZ
Z Z
is a prime ring.

Show that every nonzero homomorphic image of R= End(Vk) where V is a vector space over a division ring k is a prime ring.

Let R = End(Vk) where V is a vector space over a division ring k. Show that all ideals of R are linearly ordered by inclusion and idempotent.

Let R = End(Vk) where V is a vector space over a division ring k. Show that all ideals not equal R are prime.