Answer to Question #25254 in Abstract Algebra for Tsit Lam

Question

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.

Answer 1

First assume R is in S, where S is aleft primitive ring. Then S is a prime ring, char R = char S is either a prime
number p,or char R = 0. In the latter case, for any integer n ≥ 1, n · 1is not
a 0-divisor in S. Clearly, this implies that (R,+) is torsion-free.
Conversely, assume char R is a primep, or that (R,+) is torsion-free. In either case, R can be embedded into a
k-algebra A over some field k. Now the “left regular representation”
ϕ : A → End_k(A)
defined by ϕ(a)(b)= ab for a, b from A is anembedding of A (and hence of R) into the left (and incidentally also right)
primitive k-algebra End_k A.