Subdirectly Irreducible DQC Rings

has characterized those rings with involution J, with 1, and such that every symmetric element has an inverse under the additional hypothesis that S~~, the subring generated by ,S (the set of symmetric elements), is A, the ring itself.In this paper we raise the question as to when S~ = A under a weaker hypothesis and prove the following theorem.(A henceforth is such that 2A =A and 2x=0 implies x=0 for any xEA.)Theorem 1.Let A be a ring with 1 and suppose that S is a simple Jordan ring under the Jordan multiplication, s o t = st+ts for all s and t in S. Then either S~ = A or SEZ, the center of A, or K, the set o/ skew elements, is an ideal o/ A with K2EZ, K3 = (8), and S is an associative ring under Jordan multiplication.We note that Osborn proves that under his hypothesis, either A is a division ring; the direct sum of two division rings which are antiisomorphic, and / interchanges the summands; or 51 is a field under Jordan multiplication, K is an ideal of A where K2 = (d).A corollary to our result is Corollary 1.Let A be as in Theorem 1. Suppose /urther that A contains no nilpotent ideals.Then either A is simple or A is a direct sum o/ two simple rings which are anti-isomorphic and such that J interchanges the direct summands.

In [10], it is defined that a right (or left) ideal I of a ring R is very large if the cardinality of R/I is finite. It is also proven in [10, Theorem 3.4] that if R is a prime ring with 1 such that its characteristic is zero, then R is a right order in a simple ring with the minimum condition on one sided ideals if every large right ideal of R is very large. In the present note, we shall prove that if R is a prime ring with 1 such that its characteristic is zero and R is also a compact topological ring, then R is a right and left order in a simple ring with the minimum condition on one sided ideals, which is also a non-discrete locally compact topological ring if and only if every large right ideal of R is open. In particular, if R is an integral domain with 1 (not necessarily commutative) such that its characteristic is zero, then R is openly embeddable [13, p. 58] in a locally compact (topological) division ring if and only if every large right ideal of R is open. Following S. Warner [13], we shall say R is openly embeddable in a quotient ring Q(R) if there is a topology on Q(R) which is compatible with its structure, which induces on R its given topology and for which R is an open subset.

The aim of this paper is to obtain information about a periodic Jordan ring by using only properties of its idempotent elements. Osborn proves that a power-associative periodic ring having only one nonzero idempotent element is a division ring. so associative. He also proves that a periodic Jordan ring is a subdirect product of simple periodic Jordan rings and that a simple periodic Jordan ring is either a periodic field or a Jordan ring of capacity 2. Using these results we obtain some necessary and suficient conditions for a periodic Jordan ring to be associative, and these conditions are only given in terms of the idempotent elements. We also characterize the periodic Jordan ring which are a direct product of periodic fields and simple periodic Jordan rings of capacity two.

Let R R be a semiprime K K -algebra with unity, d d a nonzero derivation of R R , and f ( x 1 , … , x t ) f({x_1}, \ldots ,{x_t}) a monic multilinear polynomial over K K such that d ( f ( a 1 , … , a t ) ) ≠ 0 d(f({a_1}, \ldots ,{a_t})) \ne 0 for some a 1 , … , a t ∈ R {a_1}, \ldots ,{a_t} \in R . It is shown that if for every r 1 , … , r t {r_1}, \ldots ,{r_t} in R R either d ( f ( r 1 , … , r t ) ) = 0 d(f({r_1}, \ldots ,{r_t})) = 0 or d ( f ( r 1 , … , r t ) ) d(f({r_1}, \ldots ,{r_t})) is invertible in R R , then R R is either a division ring D D or M 2 ( D ) {M_2}(D) , the ring of 2 × 2 2 \times 2 matrices over D D , unless f ( x 1 , … , x t ) f({x_1}, \ldots ,{x_t}) is a central polynomial for R R . Moreover, if R = M 2 ( D ) R = {M_2}(D) , where 2 R ≠ 0 2R \ne 0 and f ( x 1 , … , x t ) f({x_1}, \ldots ,{x_t}) is not a central polynomial for D D , then d d is an inner derivation of R R .

Let A be an indiscrete locally compact primitive ring having minimal left ideals.Algebraically, A may be regarded as a dense ring of linear operators containing nonzero linear operators of finite rank on a vector space E over a division ring K, and furthermore if A is simple, then A contains only linear operators of finite rank.The topology of A induces topologies on F and A" in a natural way, so that K becomes a locally compact division ring and E a locally compact vector space over K. Theorems about locally compact division rings and locally compact vector spaces imply that if K is indiscrete, then E is necessarily finite dimensional over K, and thus A is the ring of all linear operators on a finite-dimensional vector space over a locally compact division ring.Kaplansky proved that if A has characteristic zero, then K is indeed indiscrete, but he showed by an example that E could be infinite dimensional.Kaplansky remarked [9, p. 459], however, that if A is simple, it seemed unlikely that F could be infinite dimensional, since otherwise "completeness appears to require the presence of linear transformations with infinitedimensional range."Our principal result in 1 is the verification of this conjecture; however, our argument is based not on the fact that a locally compact ring is complete, but rather on the fact that a locally compact space is a Baire space and on the symmetry between minimal left and right ideals in a simple ring.In 2 we shall present certain results about locally compact vector spaces over discrete division rings; these enable us to prove that a locally compact, indiscrete, central primitive algebra having minimal left ideals is finite dimensional provided its scalar field either is indiscrete, has characteristic zero, or is uncountably infinite.However, there is a very natural example of a locally compact, indiscrete, infinitedimensional central primitive algebra having minimal left ideals over a discrete countably infinite field of prime characteristic.1. Locally compact simple rings having minimal left ideals.We begin with a summary of the structure theory of a primitive ring A having minimal left ideals.Let e be a minimal idempotent of A, that is, an idempotent such that Ae is a minimal left ideal, or equivalently, such that eA is a minimal right ideal.Then eAe is a division ring, the additive group Ae becomes a right eAe-\tctor space by defining scalar multiplication to be the restriction of multiplication on A to

So far our emphasis has been on studying rings in terms of the module categories they admit—that is, in terms of the representations of the rings as endomorphism rings of abelian groups. As we shall see the Wedderburn Theorem for simple artinian rings can be interpreted as asserting that a ring R is simple artinian if and only if the category R M is “the same” as the category D M for some division ring D. On the other hand, if D is a division ring, then the theory of duality from elementary linear algebra asserts that the categories D FM and FM D of finitely generated left D-vector spaces and right D-vector spaces are “duals” of one another.

Given a ring R, consider the condition: (*) every maximal right ideal of R contains a maximal ideal of R. We show that, for a ring R and 0 ≠ e 2 = e ∈ R such that ele ⫋ eRe every proper ideal I of R R satisfies (*) if and only if eRe satisfies (*). Hence with the help of some other results, (*) is a Morita invariant property. For a simple ring R R[x] satisfies (*) if and only if R[x] is not right primitive. By this result, if R is a division ring and R[x] satisfies (*), then the Jacobson conjecture holds. We also show that for a finite centralizing extension S of a ring R R satisfies (*) if and only if S satisfies (*).

In a recent paper2 we have called a ring 2 primitive if 2 contains a maximal right ideal a such that the quotient (a: W) = (0). The quotient (: 1) is defined to be the largest two-sided ideal of 21 that satisfies the condition 21(a: 21) C a. Thus if 21lias an identity, (a: 21) is the largest two-sided ideal of 21 contained in a. Primitive rings appear to play a role in the general structure theory of rings analogous to that played by simple rings in the Wedderburn-Artin theory. Primitive rings are also fundamental in general representation theory since it is known that a necessary and sufficient condition that a ring 21 be primitive is that 21 be isomorphic to an irreducible ring W of endomorphisms in a commutative group T. If 2 is irreducible in T and Zi is the division ring of endomorphisms commutative with the elements of A, then R is a dense ring of linear transformations in T regarded as a vector space over Z.3 We shall call 2 a left primitive ring if 2 contains a maximal left ideal a' such that the quotient (}': 21)l = (0). Here (s': 2) 1 is the largest two-sided ideal in 2 having the property (a': 21)1 21 C a'. Evidently, a ring is left primitive if and only if it is anti-isomorphic to an irreducible ring of endomorphisms. It is an open question whether or not a primitive ring is necessarily left primitive. This is indeed the case if 2 contains minimal ideals. In many other respects the theory of primitive rings that contain minimal ideals constitutes the most satisfactory part of the theory of primitive rings. An appropriate tool for studying rings of this type is a generalization of the duality theory previously used by Dieudonn6 in studying simple rings that possess minimal ideals.4 In this note we develop this generalization of Dieudonn6's results. We also investigate certain natural topologies that can be defined in any primitive ring.

The structure of halfrings which are strong direct sums of minimal subtractive right ideals is studied.A class of right simple hemirings which contains both division hemirings and differential subsemirings of division rings is introduced and studied extensively as a tool in this investigation.A matrix representation is obtained for a class of halfrings which properly includes differential subsemirings of simple Artinian rings.1. Introduction.A semiring is a triple (H, +, ) such that (H, + ) is a semigroup with identity 0, (H, ) is semigroup with zero 0, and multiplication distributes over addition.If the addition is commutative, H is called a hemiring.A subsemiring S of a hemiring H is differential in HifVhEH 3s, t G S with s + h = t.A halfring is a hemiring H which is differential in some ring H, which is then uniquely determined.This paper continues the point of view of [9], and terminology introduced there will be extended without comment to hemirings generally.A hemiring is [right, left] simple if it has no nontrivial subtractive [right, left] ideals.It is [right, left] ideal-free if it has no nontrivial [right, left] ideals.A division semiring is a semiring with 1 = 0 all of whose nonzero elements have multiplicative inverse.It is clear that division hemirings may be characterized among nonzero hemirings as those which are right and left ideal-free, and among hemirings with identity as those which are right ideal-free.Grillet and Grillet [4], [5] have recently studied ideal-free and semisimple semirings very successfully using the techniques of semigroup theory.Earlier, Bourne [1], Bourne and Zassenhaus [2], [3], Steinfeld [8], and Wiegant [10] had studied semirings which are direct sums of minimal right ideals.The culmination of that study is an analogue of the Wedderburn theorem which may be paraphrased as follows:An ideal-free semiring with identity which is a strong direct sum of finitely many minimal right ideals is isomorphic to a full semiring of matrices over a division semiring.

A Galois theory of commutative rings

It is well known that the set of all ideals(2) of a ring forms a complete modular lattice with respect to set inclusion. The same is true of the set of all right ideals. Our purpose in this paper is to consider the consequences of imposing certain additional restrictions on these ideal lattices. In particular, we discuss the case in which one or both of these lattices is complemented, and the case in which one or both is distributive. In §1 two strictly latticetheoretic results are noted for the sake of their application to the complemented case. In §2 rings which have a complemented ideal lattice are considered. Such rings are characterized as discrete direct sums of simple rings. The structure space of primitive ideals of such rings is also discussed. In §3 corresponding results are obtained for rings whose lattice of right ideals is complemented. In particular, it is shown that a ring has a complemented right ideal lattice if and only if it is isomorphic with a discrete direct sum of quasi-simple rings. The socle [7](3) and the maximal regular ideal [5] are discussed in connection with such rings. The effect of an identity element is considered in §4. In §5 rings with distributive ideal lattices are considered and still another variant of regularity [20] is introduced. It is shown that a semi-simple ring with a distributive right ideal lattice is isomorphic with a subdirect sum of division rings. In the concluding section a type of ideal, introduced by L. Fuchs [9] in connection with commutative rings with distributive ideal lattice, and which we call strongly irreducible, is considered. Some properties of these ideals, analogous to corresponding ones for prime ideals [19], are developed. Finally, it is observed that a topology may be introduced in the set of all proper strongly irreducible ideals in such a way that the resulting space contains the spaces of prime [19] and primitive [13] ideals as subspaces. I wish to take this opportunity to thank Professor M. F. Smiley for the encouragement and many helpful suggestions he has given me throughout the preparation of this paper. 1. Some lattice-theoretic preliminaries. In this section we state two results of a strictly lattice-theoretic nature with a view toward subsequent applications to rings. Our notation and terminology is that of Birkhoff [3]. In