On the subring generated by the symmetric elements

Herstein's Lie theory revisited

Let RG denote the group ring of a group G over a commutative ring with unity R. Given a homomorphism σ:G → {± 1} and an involution ϕ of the group G, an oriented invoultion σϕ of RG is defined in a natural way. We characterize when the set of symmetric elements under this involution is a subring. This gives a unified setting for earlier work of several authors.

Let A be a simple ring which is not a 4-dimensional algebra over a field of characteristic 2, and let U be a Lie ideal of A. Then Herstein proves in [2] that either U is contained in the center Z of A, or U contains [A, A ]; if in addition U is also assumed to be an associative ring and containls [A, A], then U=-A. Herstein and Baxter also prove in [1; 3], and [4] that [A, A] mod [A, A ]CnZ is a simple Lie ring. Our object here is to give a simple proof of this fact, basing our argument only on the results of [2] quoted above, though many of Herstein's ideas will be incorporated here without explicit mention. We first recall a few definitions and make some preliminary remarks. If u and a are elements of the ring A, [u, a] will designate uac-au, and [U, A ] will denote the module generated by all [u, a] where u is in U and a in A. U is an ideal of A if U is a module and if [U, A] is contained in U. If U and V are submodules of A then U+ V is the module generated by U and V. If U is a submodule of A we shall define U:A to be the set of elements x of A such that [x, A ] C U, thus [U: A, A ] is contained in U. It is easy to check that U:A is a ring from the identity [?v, a] = [u, va] + [v, au]. When U is a Lie ideal of A then U:A is also such. The following identities will be needed in the sequel; they are easy to verify:

Let R be a primitive ring with involution *. Thus R may be considered as an irreducible ring of endomorphisms of an additive abelian group V, so that D=HomR(V, V) is a division ring. Let C be the center of D. We shall furthermore assume that CRCR. It can be shown that an involution -y--+is induced in C which has the property that 1yx = (tyx*)* for all xER. The involution * is of the first kind if -y-+-7 is the identity mapping and is of the second kind if there is a -y Oe C such that fy = -y. The set of symmetric elements of R will be denoted by S. We now assume that S satisfies a nontrivial generalized polynomial identity over C (in the sense of Amitsur). This means that there exists a nonzero element f(xl, x2, * * *, x.) in the so-called C-universal product R(x) of the C-algebra R and the free C-algebra C[X,, x2,* * *, x,, ... ] in noncommuting indeterminants xi, X2, *. * Xn, ...* such thatf(s1, S2, * * *, sn) = 0 for all SI S2, *.*.*, SnCS. For more precise details concerning the above notions we refer the reader to [1, ?4], and [2, ?3]. The usual linearization process may be used so that we may assume without loss of generality that S satisfies a nontrivial generalized (homogeneous) multilinear identity of degree n in xi, x2, ,n:

Free algebras in division rings with an involution

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 be a 2-torsion free commutative ring with involution, and δ a nonzero derivation of R. Let S be the set of symmetric elements in R, and let K be the set of anti-symmetric elements in R. In this article, we investigate the semiprimeness of the Lie rings Sδ when δ is symmetric and Kδ when δ is anti-symmetric.

Ordered ∗ -Rings

In 1996, the first author defined analogs of the concepts of complete (divisible), reduced, and periodic abelian groups, well-known in the theory of abelian groups, for arbitrary varieties of algebras. In 2021, the first author proposed a modification of the concepts of completeness and reducibility, which is more natural in the case of associative rings. The paper studies the modification of these concepts for associative rings. Artinian \(\mathbf{M}\)-complete, \(\mathbf{M}\)-reduced rings, and minimally \(\mathbf{M}\)-complete associative nilpotent rings, simple rings with unity, and finite rings are characterized.

In this paper we examine the annihilators of algebraic expressions in the symmetric, or skew-symmetric, elements of a ringR with involution. LetL be the set of elements ofR each of which is annihilated on the right by some power of each symmetric element. IfR contains no nonzero nil left ideal thenL=0. The subset ofL annihilated by a fixed power of each symmetric element is always a nil left ideal ofR of bounded index. WhenR is an algebra over a fieldF, letA be the left ideal of elements annihilated on the right by some polynomial in each symmetric element. If eitherF is uncountable or the degrees of the polynomials are bounded, thenA generates an algebraic ideal ofR. Similar results are obtained by using skew-symmetric elements instead of symmetric elements.

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

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.

CONTENTS Introduction § 0. The analytical and arithmetical hierarchies § 1. Some non-elementary logics § 2. Finite forcing § 3. Rich theories § 4. Some examples of rich theories § 5. Inverse semigroups § 6. Associative rings § 7. Division rings § 8. Open questions References

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 .