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The notion of a Γ-ring was first
introduced by Nobusawa. The class of Γ-rings contains not only all rings but also
Hestenes ternary rings. Recently, the author proved the following two theorems:
Theorem A. Let M be a Γ-ring with right and left unities and R be the right
operator ring. Then, the lattice of two-sided ideals of M is isomorphic to the
lattice of two-sided ideals of R. Theorem B. Let M be a Γ-ring such that
x ∈ MΓxΓM for every x ∈ M. If 𝒫(M) is the prime radical of the Γ-ring M, then
𝒫(Mm,n) = (𝒫(M))m,n. If a Γ-ring M has no unit elements, Theorem A is not, in
general, the case. However, it is possible to establish for any Γ-ring M, with or
without right and left unities, the result corresponding to Theorem A for a special
type of ideals, namely, prime ideals. In this note, we prove Theorem 1. The set of
all prime ideals of a Γ-ring M and the set of all prime ideals of the right
(left) operator ring R(L) of M are bijective. Applying this result to the
matrix Γn,m-ring Mm,n, we obtain Theorem 2. The prime ideals of the
Γn,m-ring Mm,n are the sets Pm,n corresponding to the prime ideals P of
the Γ-ring M, and Corollary 2. If 𝒫(M) is the prime radical of the Γ-ring
M, then 𝒫(Mm,n) = (𝒫(M))m,n. This corollary omits the assumption of
Theorem B.
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