Throughout this paper the word “ring” will mean an associative ring which
need not have an identity element. There are Artinian rings which are not
Noetherian, for example C(p∞) with zero
multiplication. These are the only such rings in that an Artinian ring
R is Noetherian if and only if R
contains no subgroups of type C(p∞)
[1, p. 285]. However, a certain class of Artinian rings is
Noetherian. A famous theorem of C. Hopkins states that an Artinian ring with
an identity element is Noetherian [3, p. 69]. The proofs of
these theorems involve the method of “factoring through the nilpotent
Jacobson radical of the ring”. In this paper we state necessary and
sufficient conditions for an Artinian ring (and an Artinian module) to be
Noetherian. Our proof avoids the concept of the Jacobson radical and depends
primarily upon the concept of the length of a composition series. As a
corollary we obtain the result of Hopkins.
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