Abstract Algebra Answers

Show that the following are equivalent:
(A) If I is a nil ideal in any ring R, then Mn(I) is nil for any n.
(B)' If I is a nil ideal in any ring, then M2(I) is nil.

Show that from Kothe’s Conjecture (“The sum of two nil left ideals in any ring is nil”.) followsthe statement:
if I is a nil ideal in any ring R, then Mn(I) is nil for any n.

Show that any ideal (resp. subring) can be realized as the kernel (resp. the image) of a ring homomorphism.?!

Let Rad R denote one of the two nilradicals, or the Jacobson radical, or the Levitzki radical of R.
For any ideal J ⊆ R such that Rad (R/J) = 0, show that J ⊇ Rad R.

Let Rad R denote one of the two nilradicals, or the Jacobson radical, or the Levitzki radical of R.
For any ideal I ⊆ Rad R, show that Rad (R/I) = (Rad R)/I.

Let Rad R denote one of the two nilradicals, or the Jacobson radical, or the Levitzki radical of R.
Show that Rad R is a semiprime ideal.

If R is commutative, then, every ideal of R is semiprime implies R is von Neumann regular.

R is von Neumann regular, prove that: every ideal I of R is idempotent.

For a ring R, prove that: if every ideal of R is semiprime, then every ideal I of R is idempotent.

Let R be a k-algebra where k is a field. Let K/k be a separable algebraic field extension.
Show that Nil*(RK) = (Nil*(R))K.