Periodic jordan rings and order structure

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.

Twenty-five years ago McCoy published a characterization of commutative subdirectly irreducible rings. This result was generalized by Thierrin to duo rings with the word “field” which appeared in McCoy's theorem replaced by “division ring”. The purpose of this note is to give another generalization in which the words “division ring” will be replaced by “simple ring with 1 ”. The techniques resemble those of McCoy and Thierrin.

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.

A celebrated theorem of Wedderburn says that a finite associative division ring is a finite (commutative) field. An elegant proof of this, due to E. Artin [3, p. 72], reduces the problem to showing the reduced (or generic) norm has a nontrivial zero, and then applies the theorem (conjectured by Artin and proved by Chevalley) that a polynomial with coefficients in a finite field and zero constant term has a nontrivial zero in that field if its degree is less than the number of variables. The Wedderburn theorem was generalized by A. A. Albert to finite strictly power-associative division rings of characteristic # 2 in [1, p. 301 ] and [2, p. i1 ] (see also [5 ]). The proof reduced the powerassociative case to the associative case by using the properties of Jordan division algebras. It is the purpose of this note to show that Artin's method carries over directly to the power-associative case using the generic norm introduced by N. Jacobson [4]. By avoiding Jordan algebras one is able to extend Albert's result to the characteristic 2 case. The usual definition of a division ring requires that left and right multiplications by a nonzero element be bijective. If the ring is finite it is necessarily an algebra, over a finite field 4, which is algebraic (even finite-dimensional) and without zero divisors. Also, as we shall see later, the algebra necessarily has a unit element. We wish to consider something slightly more general. For our purposes we say an algebraic power-associative algebra is a division algebra if it is unital and no element xO 0 is a zero divisor in the subalgebra (D[x] it generates (since 4<[x] is finite-dimensional, this is equivalent to saying x is invertible in b [x]). Our conditions are less restrictive, since the Jordan algebra of a quadratic form can be a division algebra in our sense with zero divisors. With this definition we have

When is every matrix over a division ring a sum of an idempotent and a nilpotent?

Herstein's Lie theory revisited

Let E*G be a crossed product of a division ring E and a locally indicable group G. Hughes showed that up to E*G-isomorphism, there exists at most one Hughes-free division E*G-ring. However, the existence of a Hughes-free division E*G-ring {mathcal {D}}_{E*G} for an arbitrary locally indicable group G is still an open question. Nevertheless, {mathcal {D}}_{E*G} exists, for example, if G is amenable or G is bi-orderable. In this paper we study, whether {mathcal {D}}_{E*G} is the universal division ring of fractions in some of these cases. In particular, we show that if G is a residually-(locally indicable and amenable) group, then there exists {mathcal {D}}_{E[G]} and it is universal. In Appendix we give a description of {mathcal {D}}_{E[G]} when G is a RFRS group.

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.

Some Subidempotent Radicals

Let R be a ring, Q its symmetric Martindale quotient ring, C its extended centroid, I a nonzero ideal of R and F a generalized derivation with associated non-zero derivation d of R, and fixed integers. Let be a non-zero multilinear polynomial over C in t non-commuting variables, be any subset of R and . We prove the following results:If R is prime and for all , then is central valued on R.If R is prime and , for all , then is power central valued on R, unless .If R is semiprime and for all , then , for any and , that is there exists a central idempotent element such that , d vanishes identically on eQ and is central valued on .If R is semiprime and is zero or invertible in R, for all , then either R is a division ring or it is the ring of 2 × 2 matrices over a division ring, unless when , for any and .If R is prime and I is a non-zero right ideal of R such that and , for all , then is an identity on I.Let R be prime and I a non-zero right ideal of R such that and , for all . If there exists such that , then either is power central valued on R or is an identity on I, unless .

Let R be a commutative ring with identity and let I 1 , I 2 , …, I ℓ be ideals of R . Let A denote the direct product of residue-class rings R / I k (1 ≤ k ≤ ℓ), and suppose p is a prime. We consider conditions for A to be generated as an R -algebra by a finite p -subgroup of the unit group.

In this paper, codes over the direct product of two finite commutative chain rings are studied. The standard form of the parity-check matrix is determined. The structure of self-dual codes is described. A distance preserving Gray map from the direct product of chain rings to a finite field is defined. Two upper bounds on minimum distance are obtained.

This paper is an elementary note which indicates how Harrison's primes sit in certain kinds of rings. It is proved that primes behave nicely under finite direct products. Also it is shown that any nil ideal is a subset of every prime. This gives information about the primes of artinian rings.