A note on embedding a partially ordered ring in a division algebra

Division algebras D with valuation v are studied, where D is lSnite-dimensional and totally ramified over its center F (i.e., the ramification index of v over wlF equals [D: F]). Such division algebras have appeared in some important constructions, but the structure of these algebras has not been systematically analyzed before. When vlF is Henselian a full classification of the F-subalgebras of D is given. When F has a Henselian valuation v with separably closed residue field and A is any tame central simple F-algebra, an algorithm is given for computing the underlying division algebra of A from a suitable subgroup of A*/F*. Some examples are constructed using this valuation theory, including the first example of finitbdimensional F-central division algebras D1 and D2 with D1 XF D2 not a division ring, but D1 and D2 having no common subfield K D F. Valuation theory, long a basic tool in commutative algebra, has been relatively neglected in the study of division algebras, until quite recently. Nontheless, valuations are naturally present in a number of division algebras that have been constructed to exhibit special properties, particularly algebras over iterated Laurent power series fields. For example, such division algebras have been key ingredients in Amitsur's noncrossed product construction [Am] and in Platonov's construction [P] of division algebras D with SK1(D) 7& 1. Valuations are not so prevalent on division algebras as on fields. But if a division algebra D does have a valuation, this structure contains a substantial amount of information about D which would scarcely be accessible otherwise. We consider here valued divisioll algebras D for which D is totQlly ramified and tame over its center F, i.e., for which the ramification index IrD rFI equals the dimension [D: F] of D over F and the characteristic char(D) does not divide [D: F]. (Here rD iS the value group of the valuation on D, and D is the residue division algebra. We assume throughout that [D: F] < oo.) Valued division algebras of this type appear, e.g., in Amitsur's noncrossed product paper [Am, §2], in Saltman's work on indecomposable division algebras [Sa], in certain of the MalcevNeumann division algebras considered by the first author and Amitsur [TA2,§4], etc. However, the intrinsic structure of totally ramified tame division algebras has apparently not been examined closely before. This may be because most past work on valued division algebras has concentrated on discrete valuations, when rD -Z; for such a valuation D is never totally ramified over its center F unless D = F (cf. (3.2) below). Received by the editors June 26, 1986. 1980 Mathematics Subject Clmsification (1985 Reon). Primary 16A39. 2Supported in part by F.N.R.S. 2Supported in part by the National Science Foundation. (:)1987 American Mathematlcal Society 0002-9947/87 $}.00 + $ 25 per page

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

Continuing the work in [ll],[l2] we study division algebras D = k(G) over a field k which are generated by some polycyclic-by-finite subgroup G of the multiplicative group D* of D. We discuss a specific class of examples of such division algebras that can be thought of as multiplicative analogs of the Weyl field. Furthermore, we show that the division algebras D = k(G) always contain free subalgebras of rank ≤ 2, provided G is not abelian-by-finite. Finally, we discuss some open questions concerning commutative subfields and Lie commutators in D = k(G).

We describe the associative multilinear polynomial functions over commutative integral domains. This extends Marichal and Mathonet's result on infinite integral domains and provides a new proof of Andres's classification of two-element $n$-semigroups.

Let A be a commutative integral domain with quotient field L, and let R be a subdomain of A with quotient field K. Assuming that L is a Galois extension of K, Nagata required the condition for R to be normal when A is called a Galois extension of R (see p. 31, M. Nagata, Local Rings (Wiley, New York, 1962)). However in this paper, A is considered in the case that R is not necessarily assumed to be normal. We introduce the notion of cyclic Galois extensions of integral domains and investigate several properties of such ring extensions. In particular, we completely determine the seminormalization [Formula: see text] of A in an overdomain B such that both A ⊆B are cyclic Galois extensions of R.

In this paper, we present an isomorphism between the ring of general polynomials over a division algebra D with center F and the group ring of the free monoid with [D : F ] variables over D. Using this isomorphism, we define the characteristic polynomial of any matrix over any division algebra, i.e., a general polynomial with one variable over the algebra whose roots are precisely the left eigenvalues.Furthermore, we show how the left eigenvalues of a 4 × 4 quaternion matrices can be obtained by solving a general polynomial equation of degree 6.

An elementary proof of the first fundamental theorem of vector invariant theory

We investigate one-dimensional ‘generalized convolution behaviors’ (gen. beh.) that comprise differential and delay-differential behaviors in particular. We thus continue work of, for instance, Brethé, van Eijndhoven, Fliess, Gluesing-Luerssen, Habets, Loiseau, Mounier, Rocha, Vettori, Willems, Yamamoto, Zampieri of the last twenty-five years. The signal space for these behaviors is the space E of smooth complex-valued functions on the real line. The ring of operators is the commutative integral domain E′ of distributions with compact support with its convolution product that acts on E by a variant of the convolution product and makes it an E′-module. Both E and E′ carry their standard topologies. Closed E′-submodules of finite powers of E were introduced and studied by Schwartz already in 1947 under the name ‘invariant varieties’ and are called gen. beh. here. A gen. beh. is called a behavior if it can be described by finitely many convolution equations. The ring E′ is not noetherian and therefore the standard algebraic arguments from one-dimensional differential systems theory have to be completed by methods of topological algebra. Standard constructions like elimination or taking (closed) images of behaviors may lead to gen. beh. and therefore the consideration of the latter is mandatory. It is not known whether all gen. beh. are indeed behaviors, but we show that many of them are, in particular all autonomous ones. The E′-module E is neither injective nor a cogenerator and, in particular, does not admit elimination in Willems’ sense. But the signal submodule PE of all polynomial-exponential signals is injective for finitely generated modules and thus admits elimination. This is a useful replacement and approximation of the injectivity of E since the polynomial-exponential part of any gen. beh. is dense in it. We also describe a useful replacement of the cogenerator property and thus establish a strong relation between convolution equations and their solution spaces. Input/output structures of gen. beh. exist and are used to prove that also many nonautonomous generalized behaviors are indeed behaviors. The E′-torsion elements of E, i.e., the smooth functions which satisfy at least one nonzero convolution equation, are called ‘mean-periodic functions’ and were studied by many outstanding analysts. Their results are significant for gen. beh.

Let R be a commutative integral domain with 1. It is trivial to see that in the polynomial ring R[X], any nonconstant monic polynomial can be factored into a product of nonconstant monic polynomials which are irreducible in R[X]. However, in an arbitrary domain, such a factorization need not be unique. We show uniqueness occurs exactly when R is integrally closed.

In this paper we extend classical results of the invariant theory of finite groups to the action of a finite-dimensional semisimple Hopf algebra H on a special algebra A, which is homomorphically mapped onto a commutative integral domain, and the kernel of this map contains no nonzero H-stable ideals. We prove that the algebra A is finitely generated as a module over a subalgebra of invariants, and the latter is finitely generated as a k-algebra. We give a counterexample to the finite generation of a non-semisimple Hopf algebra.

A torsion-free module M over a commutative integral domain R is said to be LCM-stable over R if (Ra ∩ Rb)M = Ma ∩ Mb for all a, b ∈ R. We show that if the module M is LCM-stable over a GCD-domain R, then the polynomial module M[X] is LCM-stable over R[X]; if R is a w-coherent locally GCD-domain, then LCM-stability and reflexivity are equivalent for w-finite type torsion-free R-modules. Finally, we introduce the concept of w-LCM-stability for modules over a domain. Then we characterize when the module M is w-LCM-stable over the domain in terms of localizations and t-Nagata modules, respectively. Also we characterize Prüfer v-multiplication domains and Krull domains in terms of w-LCM-stability.

Let R be a commutative integral domain with field of fractions F and let Q be a finite-dimensional central simple F-algebra. If R is a Prufer domain then it is still unknown whether or not R can be extended to a Prufer order in Q in the sense of Alajbegovic and Dubrovin (J. Algebra, 135: 165–176, 1990). In this paper we investigate a more general class of rings which we call rings of Prufer type and we will prove an extension theorem for these rings. Under special assumptions this result also leads to an extension theorem for certain Prufer domains.

Let R be a prime ring and T be a nontrivial automorphism of R. If xxT—xTx is in the center of the ring for every x in R, then R is a commutative integral domain.

We consider some generalizations of the notion of divisible module, and study when our “divisible” modules coincide with injective modules. Here the base ring $$R$$ is usually a (not-necessarily commutative) integral domain. For instance, we prove that an integral domain $$R$$ is right noetherian and right hereditary if and only if the class of injective right $$R$$ -modules coincides with the class of $$h$$ -divisible right $$R$$ -modules, where “ $$h$$ -divisible” is the non-commutative version of the corresponding notion of $$h$$ -divisible module introduced by Matlis for modules over commutative domains. We then define and study a divisible right $$R$$ -module $$\Delta $$ , which is the non-commutative analogue of the tilting module $$\partial $$ defined by L. Fuchs. The case we are mainly interested in is that of the finitely generated free algebra $$R:=k\langle x_1,x_2,\ldots ,x_n\rangle $$ over a commutative field $$k$$ , where $$x_1,x_2,\ldots ,x_n$$ are non-commuting indeterminates over $$k$$ . For $$n\ge 2$$ , this $$R$$ is a non-noetherian fir, hence, in particular, it is a hereditary integral domain.

We use a variation on a construction due to Corner 1965 to construct (Abelian) groups A that are torsion as modules over the ring End (A) of group endomorphisms of A. Some applications include the failure of the Baer-Kaplansky Theorem for Z[X]. There is a countable reduced torsion-free group A such that IA = A for each maximal ideal I in the countable commutative Noetherian integral domain, End (A). Also, there is a countable integral domain R and a countable. R-module A such that (1) R = End(A), (2) T0 ⊗RA ≠ 0 for each nonzero finitely generated (respectively finitely presented) R-module T0, but (3) T ⊗RA = 0 for some nonzero (respectively nonzero finitely generated). R-module T.