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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.
In Progress...
Show that R' =
Z nZ
0 Z
is not prime ring.
In Progress...
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
In Progress...
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.
In Progress...
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.
In Progress...
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.
In Progress...
Show that a ring R is semiprime iff, for any two ideals A,B in R, AB = 0 implies that A ∩ B = 0.
In Progress...
Let A,B be left (resp. right) ideals in a semiprime ring R. Show that AB = 0 iff BA = 0.
In Progress...
Let A ideal in a semiprime ring R. Show that annr(A) = annl(A).
In Progress...
Show that, with respect to inclusion, the set S of semiprime ideals in any ring R forms a lattice having a smallest element and a largest element.
In Progress...
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