Vol. 98, No. 2, 1982

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Prime ideals in gamma rings

Shoji Kyuno

Vol. 98 (1982), No. 2, 375–379

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.

Mathematical Subject Classification
Primary: 16A78, 16A78
Received: 20 June 1980
Revised: 15 September 1980
Published: 1 February 1982
Shoji Kyuno