Vol. 18, No. 3, 1966

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On the Γ-rings of Nobusawa

Wilfred Eaton Barnes

Vol. 18 (1966), No. 3, 411–422

N. Nobusawa recently introduced the notion of a Γ-ring, more general than a ring, and obtained analogues of the Wedderburn theorems for Γ-rings with minimum condition on left ideals. In this paper the notions of Γ-homomorphism, prime and (right) primary ideals, m-systems, and the radical of an ideal are extended to Γ-rings, where the defining conditions for a Γ-ring have been slightly weakened to permit defining residue class Γ-rings. The radical R of a Γ-ring M is shown to be an ideal of M, and the radical of M∕R to be zero, by methods similar to those of McCoy. If M satisfies the maximum condition for ideals, the radical of a primary ideal is shown to be prime, and the ideal QM is P-primary if and only if Pn Q for some n, and AB Q, A⊈P implies B Q. Finally, in Γ-rings with maximum condition, if an ideal has a primary representation, then the usual uniqueness theorems are shown to hold by methods similar to those of Murdoch.

Mathematical Subject Classification
Primary: 16.96
Received: 19 January 1965
Published: 1 September 1966
Wilfred Eaton Barnes