Descending Chain Condition Research Articles

The Double Chain Condition in Cyclic Operator Groups

Naturally one wonders whether condition (b) is a consequence of (a) and added strength is given to such a conjecture, if one remembers Hopkins' 3 Theorem to the effect that in rings possessing an identity the ascending chain condition for right-ideals is a consequenice of the descending chain condition. However, it is possible to construct examples of cyclic groups where the descending, though not the ascenidinig, chain conditioni is satisfied by the admissible subgroups. In the light of Hopkins' Theorem, just quoted, it is only natural to assume that there will exist a large class of groups where cyclicity and descending chain condition imply the ascending chain condition; and it is the object of the present note to exhibit such classes of groups. Let A be an abelian group where the composition is written as addition a + b; and assume that A admits the elements in the system M as operators (= right multipliers). The M-subgroup S of A is said to be of length n = n(S), if the M-group S possesses a compositioni series 4 of length n. An ill-subgroup J of A is termed minimal, if Jt# 0 and if J does not

A characterization of semi-simple rings with the descending chain condition

A characterization of semi-simple rings with the descending chain condition

A Topology for the Set of Primitive Ideals in an Arbitrary Ring

1. In a recent paper we have called a ring ? primitive if a contains a maximal right ideal a whose quotient 3: P2 = 0.1 In general if a is any right ideal 3: W is the totality of elements b such that xb e S for all x in 1. 3: 2t is a two-sided deal and if 21 has an identity, : W1 is the largest twosided ideal of W1 contained in '. The primitive rings appear to play the same r61le in the general structure theory of rings that is played by simple rings in the classical theory of rings that satisfy the descending chain condition for one-sided ideals. Corresponding to the Wedderburn-Artin structure theorem on simple rings satisfying the descending chain condition we have the theorem that if 21 is a primitive ring 7 0, W1 is isomorphic to a dense ring of linear transformations in a suitable vector space over a division ring. W e shall call a two-sided ideal e3 in W1 a primitive ideal if Wg ^ 21 and 21 !3 is a primitive ring. It is known that if 82 is not a radical ring then 21 contains primitive ideals. Moreover, in this case the intersection II3 of all the primitive ideals in 2t coincides with the radical T of 91. In particular if 1 is semi-simple, II3 = 0. If 21 is a ring with an identity, 21 is not a radical ring. Hence if (S is any two-sided ideal 7 21 in W1, W1 contains a primitive ideal 3 6. Since W2 e3 (21 S) (3 (), e3 is primitive in 21. Thus if 21 is a ring with an identity any two-sided ideal E # 1 can be imbedded in a primitive ideal. Any commutative primitive ring is a field. Consequently any primitive ideal in a commutative ring is maximal. On the other hand, in a noncommutative ring there may exist primitive ideals that are not maximal. For example, the ring 8 of all linear transformations in an infinite dimensional vector space 9a over a division ring is primitive. Hence (0) is a primitive ideal. However, 8 contains as a proper two-sided ideal the set a of finite valued linear transformations in T9.

Structure theory of simple rings without finiteness assumptions

It is the purpose of this paper to lay the foundations of a general theory of simple rings, both associative and non-associative. In part I we obtain the structure of simple associative rings that either contain minimal right ideals or contain maximal right ideals. In either case we obtain a realization of our ring 2f as a certain type of ring of linear transformations in a vector space over a division ring. If 2{ has a minimal right ideal this realization is essentially unique and we can prove a converse theorem that rings of linear transformations having certain properties are simple and contain minimal right ideals. Thus the theory here is essentially as complete as that of the classical case of rings that satisfy the descending chain condition for right ideals. Our structure theory hinges on a general theorem (Theorem 6) on the structure of irreducible rings of endomorphisms of commutative groups. This theorem may be regarded as an extension of Burnside's theorem on irreducible algebras of matrices. The classical theory of simple rings with descending chain condition is a simple consequence of our results and we believe that the present treatment is more transparent than the methods of proof previously given(1) . In studying an arbitrary non-associative ring 9t one is led to consider the associative ring 92 generated by the left and the right multiplications x->ax and x-xcxa acting in 2. If 21 is simple, 91 is an irreducible ring of endomorphisms. In part II we describe the structure of $1 in terms of the multiplication centralizer ( of 2[ defined to be the totality of endomorphisms 'y in 2 such that (xy)y=(xy)y=x(yy). We define the concepts of center, central algebra and extension of the underlying field of an algebra. In Part III we investigate a special type of simple associative algebra that may be regarded as a generalization of the concept of the complete algebra of linear transformations in a vector space over a field, or equivalently, of the concept of the complete matrix algebra. There are many points of contact between the discussion here (and in other parts of the paper) and recent work in the theory of rings of transformations in Banach spaces and in vector spaces over the fields of reail and complex numbers(2).

The Theory of Simple Rings

1. Vector spaces over rings. If R is a ring, we call a set V of elements an R-vector space if its elements form an abelian group under addition and if R is an operator domain on this abelian group, satisfying the usual axiomsnamely the distributive law and the condition that multiplication in the ring is the same as combination of operators. Usually R operates on the left side, and in that case the product of operators corresponds to the function-theoretic product (0-T (x) = o(r (x))) ; but occasionally we write the operators on the other side, and then this is not true. In the following, we shall usually assume that no non-zero vector is annihilated by every element of R. The only spaces we shall consider will be R-subspaces. A space is said to satisfy the minimal condition if any set of R-subspaces contains a smallest one. This is equivalent to the descending chain condition; that is, to the condition that every descending chain of subspaces must be finite. The minimal condition on spaces is occasionally easy to verify by the following: