Injective modules and divisible modules over hereditary rings

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.

Injective modules and semistar operations

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.

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.

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.

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.

A nontrivial locally trivial algebra

Pure-injective and RD-injective R-modules over domains R have been investigated by many authors. We introduce another class of R-modules, called weak-injective modules, which turn out to be useful in addressing several unanswered questions between the two classes of modules. We also find that this class is an envelope class over any domain, giving a partial answer to the existence of envelope classes in the hierarchy of injective and divisible modules. Communicated by I. Swanson.

Let R be a commutative integral unital domain and L a free noncommutative Lie algebra over R. In this article we show that the ring R and its action on L are 0-interpretable in L, viewed as a ring with the standard ring language $+ , \cdot ,0$. Furthermore, if R has characteristic zero then we prove that the elementary theory $Th\left( L \right)$ of L in the standard ring language is undecidable. To do so we show that the arithmetic ${\Bbb N} = \langle {\Bbb N}, + , \cdot ,0\rangle $ is 0-interpretable in L. This implies that the theory of $Th\left( L \right)$ has the independence property. These results answer some old questions on model theory of free Lie algebras.

In the paper [Marichal, J.-L. and Mathonet, P., A description of n-ary semigroups polynomial-derived from integral domains, Semigroup Forum, 83 (2011), No. 2, 241–249] the authors provide a complete classification of the nsemigroups, defined by polynomial functions over infinite commutative integral domains with identity (i.e., the n-semigroup structures polynomial derived from integral domains). We remark that some results from that paper can be extended for the n-semigroups polynomial-derived defined over infinite semidomains with zerosumfree element. Furthermore, in this note we give a description of (n, m)- semiring structures defined by polynomial functions over such semidomains.

This note is devoted to the study of divisible modules over arbitrary commutative domains R with 1. Whereas divisible modules over Dedekind domains can be completely characterized by numerical invariants, not much is known about their structures in the general case. Divisible modules have been studied by Matlis [6] who was the first to distinguish between divisibility in general and h-divisibility. He established an important duality between h-divisible torsion modules and complete torsion-free R-modules [7] . Matlis [6] and Hamsher [4] characterized those domains R for which all divisible R-modules are h-divisible, e.g. by the property that the field Q of quotients of R has projective dimension 1.

If (R, M) and (S, N) are quasilocal (commutative integral) domains and f : R → S is a (unital) ring homomorphism, then f is said to be a strong local homomorphism (resp., radical local homomorphism) if f(M) = N (resp., f(M)⊆N and for each x ∈ N, there exists a positive integer t such that xt ∈ f(M)). It is known that if f : R → S is a strong local homomorphism where R is a pseudovaluation domain that is not a field and S is a valuation domain that is not a field, then f factors via a unique strong local homomorphism through the inclusion map iR from R to its canonically associated valuation overring (M : M). Analogues of this result are obtained which delete the conditions that R and S are not fields, thus obtaining new characterizations of when iR is integral or radicial. Further analogues are obtained in which the “pseudovaluation domain that is not a field” condition is replaced by the APVDs of Badawi‐Houston and the “strong local homomorphism” conditions are replaced by “radical local homomorphism.”

Let R be an integral domain, K its field of fractions, and M n (R) the R-algebra of square matrices of order n and entries from R. In this paper we present two ways of constructing space-time block codes as submodules of M n (R) for n ≥ 1: first, by embedding free associative R-algebras of finite rank n with no zero divisors into M n (R); second, by injecting free R-modules of finite rank n into M n (R) ∩ GL n (K) ∪{ 0}. Some examples of such space-time block codes are given.

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.