The notion of a Γring was first
introduced by Nobusawa. The class of Γrings contains not only all rings but also
Hestenes ternary rings. Recently, the author proved the following two theorems:
Theorem A. Let M be a Γring with right and left unities and R be the right
operator ring. Then, the lattice of twosided ideals of M is isomorphic to the
lattice of twosided ideals of R. Theorem B. Let M be a Γring such that
x ∈ MΓxΓM for every x ∈ M. If 𝒫(M) is the prime radical of the Γring M, then
𝒫(M_{m,n}) = (𝒫(M))_{m,n}. If a Γring M has no unit elements, Theorem A is not, in
general, the case. However, it is possible to establish for any Γring 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 Γring 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 Γ_{n,m}ring M_{m,n}, we obtain Theorem 2. The prime ideals of the
Γ_{n,m}ring M_{m,n} are the sets P_{m,n} corresponding to the prime ideals P of
the Γring M, and Corollary 2. If 𝒫(M) is the prime radical of the Γring
M, then 𝒫(M_{m,n}) = (𝒫(M))_{m,n}. This corollary omits the assumption of
Theorem B.
