Type Radicals

We show that the Kauffman bracket skein module (KBSM) for small Seifert fibered [Formula: see text]-manifolds with singular fibers of multiplicity two is a free module of rank [Formula: see text], where [Formula: see text] is the Euler number, and we construct its basis. This family of [Formula: see text]-manifolds includes as a special case the quaternionic [Formula: see text]-manifold for which KBSM was first computed by J. Harris.

Bireflectionality of the orthogonal group over a valuation domain

Let E and G be free modules of rank e and g, respectively, over a commutative noetherian ring R. The identity map on E*⊗ G induces the Koszul complex and its dual Let H𝒩(m,n,p) be the homology of the top complex at Sym m E*⊗ Sym n G⊗ ∧ p (E*⊗ G) and Hℳ(m,n,p) the homology of the bottom complex at D m E⊗ D n G*⊗ ∧ p (E⊗ G*). It is known that H𝒩(m,n,p)≅ Hℳ(m′, n′, p′), provided m + m′ = g − 1, n + n′ = e − 1, p + p′ = (e − 1)(g − 1), and 1 − e ≤ m − n ≤ g − 1. In this article, we exhibit a complex 𝕐 and explicit quasi-isomorphisms from 𝕐 to two complexes, as described above, for the appropriate choice of parameters, which give rise to this isomorphism. Our quasi-isomorphisms may be formed over the ring of integers; they can be passed to an arbitrary ring or field by base change. All of our work is equivariant under the action of the group GL(E) × GL(G); that is, everything we do is independent of the choice of basis. Knowledge of the homology of the top complex is equivalent to knowledge of the modules in the resolution of the Segre module Segre(e,g, ℓ), for ℓ = m − n. The modules {Segre (e,g, ℓ)|ℓ ∈ ℤ} are a set of representatives of the divisor class group of the determinantal ring defined by the 2 × 2 minors of an e × g matrix of indeterminates. If R is the ring of integers, then the homology H𝒩(m,n,p) is not always a free abelian group. In other words, if R is a field, then the dimension of H𝒩(m,n,p) depends on the characteristic of R. The module H𝒩(m,n,p) is known when R is a field of characteristic zero; however, this module is not yet known over arbitrary fields. The modules in the minimal resolution of the universal ring for finite length modules of projective dimension two are equal to modules of the form H𝒩(m,n,p).

Let $V$ be a submodule of a direct sum of some elements in $\mathcal{U}$, and $X$ be a submodule of a direct sum of some elements in $\mathcal{N}$, where $\mathcal{U}$ and $\mathcal{N}$ are families of $R$-modules. A $\mathcal{U}$-free module is a generalization of a free module. According to the definition of $\mathcal{U}$-free module, we define three kinds of projective$_{\mathcal{U}}$ in this research, i.e., projective$_{\underline{\mathcal{U}}}$, projective$_{\mathcal{U}}$ module, and strictly projective$_{\mathcal{U}}$ module. The notion of strictly projective$_{\mathcal{U}}$ is a generalization of the projective module. In this research, we discuss the relationship between projective modules and the three types of modules. Furthermore, we show that the properties of $\mathcal{U}$ impact the properties of the projective$_{\mathcal{U}}$ module so that we can determine some properties of the projective$_{\mathcal{U}}$ module based on the properties of the family of $\mathcal{U}$ of $R$-modules.

This paper is a continuation of [D. Adamović and G. Radobolja, Free field realization of the twisted Heisenberg–Virasoro algebra at level zero and its applications, J. Pure Appl. Algebra 219(10) (2015) 4322–4342]. We present certain new applications and generalizations of the free field realization of the twisted Heisenberg–Virasoro algebra [Formula: see text] at level zero. We find explicit formulas for singular vectors in certain Verma modules. A free field realization of self-dual modules for [Formula: see text] is presented by combining a bosonic construction of Whittaker modules from [D. Adamović, R. Lu and K. Zhao, Whittaker modules for the affine Lie algebra [Formula: see text], Adv. Math. 289 (2016) 438–479; arXiv:1409.5354] with a construction of logarithmic modules for vertex algebras. As an application, we prove that there exists a non-split self-extension of irreducible self-dual module which is a logarithmic module of rank two. We construct a large family of logarithmic modules containing different types of highest weight modules as subquotients. We believe that these logarithmic modules are related with projective covers of irreducible modules in a suitable category of [Formula: see text]-modules.

In this paper we take a natural homomorphism f M : U Hom R ( M , N ) → HomU ( UM , U N ) where U is an R-Algebra , R=CenU and M , N are any two R-modules.Here we have shown that 1. If P is any projective module then f P is a monomorphism. 2. If G be any generator then f ) ( n G is an epimorphism. 3. If M be a generator and N is any free module then f ) (M TrN is always a monomorphism. 4. If M is a generator and N is any finitely generated free R-module then f ) (M TrN is an isomorphism. 5. If M be any artinian module such that it is subdirect product of simple module then f M is monomorphism iff for each submodule K , f K is monomorphism. 6. Let M be a generator and N is any free module if U is finitely presented then f ) (M TrN is an isomorphism. 7. Let M be any left R-module such that f ) ( Re / N j M M is an isomorphism then f M is an isomorphism.

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.

On direct sums of Baer modules

The trend in large body, high performance integrated circuit packaging for the 32 nm semiconductor node and beyond is towards low dielectric loss to enable high bandwidth / low loss channels, and low thermal expansion to protect fragile, ultra-low dielectric constant (k) chip dielectric materials from differential expansion stress. A low coefficient of thermal expansion (CTE), low dielectric loss laminate composite was developed using industry standard Sequential Build Up (SBU) fabrication techniques and novel laminate materials. This laminate technology was used in assembly of a Flip Chip Plastic Ball Grid Array (FC PBGA) module including a silicon test structure developed for 32 nm Custom Logic development. The same silicon test structure and laminate design were also used to fabricate modules using conventional high volume laminate materials. Various laminate physical parameters including composite CTE were determined. The warpage shape of each laminate was initially characterized at room temperature, and over the temperature range from 25°C to 240°C. Warpage of critical package features was measured and tracked throughout the various steps of the lead free flip chip module assembly process for multiple laminate cross sections, core and buildup materials. A quantity of assemblies of each type was built and measured, data is reported. Advantages and disadvantages of each laminate module type and implications for robust package assembly as evidenced by these results are discussed.

ABSTRACTLet F1 be an equivalence of type I between two categories of modules with hermitian forms. In other words, suppose F is an equivalence between the two underlying categories of modules, and F1 is an equivalence, given by and F1(f) = F(f), such that the non-singularity of a free hyperbolic module of hyperbolic rank 1 is preserved by F1. Then every equivalence generated by a set of hermitian Morita equivalence data is of type I and every equivalence of type I arises from a set of hermitian Morita equivalence data.

The definition of the determinant of an endomorphism of a free module depends on the following fact: If F is a free R-module of rank n, then the homogeneous component ∧nF, of degree n, of the exterior algebra ∧ F of F is a free R-module of rank one. If a is an endomorphism of F, then a extends to an endomorphism of ∧ F which in ∧nF is therefore multiplication by an element of R. That factor is then defined to be the determinant of α. (A discussion of this theory may be found in [4].)This procedure cannot be applied in general to finitely generated projective modules since, for such modules, it may happen that no homogeneous component of the exterior algebra is free of rank one.

Given a ring R, it is known that the topological space BGl(R)+ is an infinite loop space. One way to construct an infinite loop structure is to consider the category F of free R-modules, or rather its classifying space BF, as food for suitable infinite loop space machines. These machines produce connective spectra whose zeroth space is (BF)+ = ZXBG1(R)+. In this paper we consider categories Co(F) = F, C1(F),... of parametrized free modules and bounded homomorphisms and show that the spaces (BCo)+ = (BF)+, (BC1)+,... are the connected components of a nonconnective ω-spectrum BC(F) with πiBC(F) = Ki(R) even for negative i.

In this paper, we study the dimension of a module over a commutative ring, which is defined to be the length of a longest chain of prime submodules. This notion is analogous to the usual Krull dimension of a ring. We investigate how some bounds on the dimension of modules are related to the structure of the underlying ring. The dimension of finitely generated modules over a Dedekind domain is also studied. By examining the structure of prime submodules, a formula for the dimension of a free module of finite rank, over a Noetherian one-dimensional domain, is obtained.

In this work we study flat modules over commutative noetherian rings under two kinds of restriction: that the modules are either submodules of free modules or that they have finite rank. The first ones have nicely behaved annihilators: they are generated by idempotents. Among the various questions related to flat modules of finite rank, emphasis is placed on discussing conditions implying its finite generation, as for instance, (i) over a local ring, a flat module of constant rank is free, and (ii) a flat submodule of finite rank of a free module is finitely generated. The rank one flat modules already present special problems regarding its endomorphism ring; in a few cases it is proved that they are flat over the base ring. Finally, a special class of flat modules—unmixed—is discussed, which have, so to speak, its source of divisibility somewhat concentrated in the center of its endomorphism ring and thus resemble projective modules over flat epimorphic images of the base ring.

Multiple zeta functions and asymptotic structure of free Abelian groups of finite rank