Question #25254

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.

Expert's answer

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.

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

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

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