Abstract Algebra Answers

Let R be a ring which acts faithfully and irreducibly on a left module V . Let v ∈ V and A be a nonzero right ideal in R. Show that A • v = 0 ⇒ v = 0.

Which of the following implications are true? R left primitive ⇐⇒ Mn(R) left primitive.

Let R be a left primitive ring. Show that for any nonzero idempotent e ∈ R, the ring A = eRe is also left primitive

Show that a ring R can be embedded into a left primitive ring iff either char R is a prime number p > 0, or (R, +) is a torsion-free abelian group.

Show that from: rad (R[t]) = (Nil*R)[t] follows Kothe’s Conjecture. (“The sum of two nil left ideals in any ring is nil”.)

Show that from if I is a nil ideal in any ring R, then I[t] ⊆ rad R[t], follows rad (R[t]) = (Nil*R)[t] for any ring R.

Show that from: if I is a nil ideal in any ring R, then Mn(I) is nil for any n, follows: if I is a nil ideal in any ring R, then I[t] ⊆ rad R[t].

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.