A Characterization of Semi-simple Rings with the Descending Chain Condition

H. Weyl has defined a semi-simple algebra (of finite rank) to be an algebra which admits a faithful semi-simple linear representation. Now, algebras are rings with a field of operators; Artin and others have shown that the theory of semi-simple algebras can be generalized to a theory of semi-simple rings (without the field of operators) provided we replace the condition of finite rank by suitable finiteness conditions. (Both the ascending and descending chain conditions were assumed, but it was later shown that the descending chain condition was sufficient.) The notion of semi-simplicity is defined by the condition that the radical reduces to {o}, there being several equivalent definitions of the radical. We introduce another one below. The question now arises whether the Weyl definition could not be extended to the case of rings. To do this, we must extend to an arbitrary ring the notion of a linear representation of an algebra. This can be done by replacing the consideration of the algebra of matrices by the more general notion of the ring of endomorphisms of an abelian group : a representation of a ring A will be a homomorphism p of A into the ring of endomorphisms of an additive group S0Î. Let such a representation be given; we can define a law of composition, (a, m) —>am, between elements of A and of 3D? by writing am= |p(a)} (m). The composite object formed by 93? and this law of composition is called an A-module, A sub-module of an A-module 9ft is a subset 9i of SDÎ which is a subgroup of the additive group of SD? and is such that AyiQyi. (AW is defined to be the set of all finite sums]T)$-a<m<, a<£-4, mi&St.) A homomorphism of an A -module SDÎ into an A -module $R' is a homomorphism h of the additive group of SDÎ into the additive group of 5DÎ' which is such that h(am)=ah(m) for all aÇ^A, mGW:. An A -module S0Î is said to be simple if its only submodules are {o} and itself. Concerning simple modules, we have the well known lemma: