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 Q≠M 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.
