The notion of a Γ-ring was
introduced by N. Nobusawa. The class of Γ-rings contains not only all rings but also
Hestenes ternary rings. Recently, W. E. Barnes, J. Luh, W. E. Coppage and the
author studied the structure of Γ-rings and obtained various generalizations
analogous of corresponding parts in ring theory. The object of this paper is to study
the properties of prime Γ-rings. Main results are the following theorems: (1) A Γ-ring
M is a subdirect sum of prime Γ-rings if and only if 𝒫(M) = 0, where 𝒫(M)
denotes the prime radical of M. (2) For the matrix Γn,m-ring Mm,n we have
𝒫(Mm,n) = (𝒫(M))m,n, where M is a ring such that x ∈ MΓxΓM for every
x ∈ M.
