This paper deals with the unit groups of the endomor- phism rings of projective modules over polynomial rings and further over formal power series rings. A normal subgroup of the unit group is dened and discussed. The local splitting properties of elements of endomorphism rings of projective modules over polynomial rings are given. So, if P is a projective module over a ring A, then (id+ xEndA(x)(P(x))) is constructed. This is a normal subgroup of the unit group of endomorphism rings of a projective module over a poly- nomial ring. If (A,m) is a local ring with dim(A) ≥ µ(m), then we show that (id+ xEndA(x)(P(x))) is a normal subgroup of SLA(x)(P(x)). Also, we will show that under these conditions the similar conclusion can be drawn for the ring of formal power series. In section 3, let P be a projective A-module. Let s1, s2 ∈ A be such that As1 + As2 = A. Then we use the splitting lemma of Quillen to show that every element of (idPs1s2 + xEnd As1s2(x) (Ps1s2 (x))) has two decompositions. Finally, we generalize the result to Theorem 3.4.

We research properties of a corresponding Cauchy theorem of hyperholomorphic functions in an open set of product complex spaces, in the sense of complex Clifford analysis.

There are several ways to normalize a Haar measure on the orders in a quaternion algebra. In order to understand the arithmetic properties of orders, we study the volumes of orders and compute them for some cases.

It is an interesting problem to study fixed points of an element in the holonomy group of an affine manifold. We compute the limit of a sequence of projective transformations and verify relations between fixed points and kernels.

In this note we consider the hyponormality of Toeplitz operators <TEX>$T_{\varphi}$</TEX> on the Weighted Bergman space <TEX>$A^2_{\alpha}(\mathbb{D})$</TEX> with symbol in the class of functions <TEX>$f+\bar{g}$</TEX> with polynomials <TEX>$f$</TEX> and <TEX>$g$</TEX> of degree 2.

In the present paper, we evaluate the analytic conditional Fourier-Feynman transforms and convolution products of unbounded function which is the product of the cylinder function and the function in a Banach algebra which is defined on an analogue o Wiener space and useful in the Feynman integration theories and quantum mechanics. We then investigate the inverse transforms of the function with their relationships and finally prove that th analytic conditional Fourier-Feynman transforms of the conditional convolution products for the functions, can be expressed in terms of the product of the conditional Fourier-Feynman transforms of each function.

The notion of vague -ideals of BCI-algebras is introduced, and several properties of them are investigated. Relations between a vague ideal and a vague -ideal are discussed. Characterizations of a vague -ideal are considered.

Abstract. In this paper, we give some properties on submoduletransforms. 0. IntroductionLet M be a module over commutative ring Rwith identity, Stheset of nonzero divisors of Rand R S the total quotient ring of R. For anonzero ideal Iof R, let I 1 = fx2R S jxIRg. Iis said to be aninvertible ideal of Rif II 1 = R. Put T= ft2Sjtm= 0 for some m2Mimplies m= 0g. Then Tis a multiplicatively closed subset of Sand ifMis torsion free, then T= S([9,Proposition 1.1]). In particular, if Misa faithful multiplication module then Mis torsion free ([4,Lemma 4.1])and so T= S. So in this case, R T = R S . Let Nbe a submodule of M. Ifx= rt 2R T and n2N, then we say that xn2Mif there exists m2Msuch that tm= rn. Then this is a well de ned operation([9,p399]). Fora submodule N of M, N 1 = fx2R T jxN Mg= [M : R T N]. Wesay that Nis invertible in Mif NN 1 = Mand Mis called a Dedekind(resp. Pruf er) module providing that every nonzero (resp. every nonzero nitely generated) submodule of Mis invertible.Mis called a multiplication module if every submodule Nof Mhasthe form IM for some ideal I of R. An R module M is said to befaithful if Ann (M) = [0 :

Let <TEX>$\mathcal{L}$</TEX> be a subspace lattice on a Hilbert space <TEX>$\mathcal{H}$</TEX>. And let X and Y be operators acting on a Hilbert space <TEX>$\mathcal{H}$</TEX>. Let <TEX>$sp(x)=\{{\alpha}x\;:\;{\alpha}{\in}\mathcal{C}\}$</TEX> <TEX>$x{\in}\mathcal{H}$</TEX>. Assume that <TEX>$\mathcal{H}=\overline{range\;X}{\oplus}sp(h)$</TEX> for some <TEX>$h{\in}\mathcal{H}$</TEX> and < <TEX>$h$</TEX>, <TEX>$E^{\bot}Xf$</TEX> >= 0 for each <TEX>$f{\in}\mathcal{H}$</TEX> and <TEX>$E{\in}\mathcal{L}$</TEX>. Then there exists an operator A in Alg<TEX>$\mathcal{L}$</TEX> such that AX = Y if and only if <TEX>$sup\{\frac{{\parallel}E^{\bot}Yf{\parallel}}{{\parallel}E^{\bot}Yf{\parallel}}\;:\;f{\in}H,\;E{\in}\mathcal{L}\}$</TEX> = K < <TEX>${\infty}$</TEX>. Moreover, if the necessary condition holds, then we may choose an operator A such that AX = Y and <TEX>${\parallel}||A{\parallel}=K$</TEX>.