Real <Emphasis Type="Italic">AW</Emphasis><Superscript>*</Superscript>-Algebras of type I

Let R be a real AW ^*-algebra, and suppose that its complexification M = R + iR is also a (complex) AW ^*-algebra. We prove that R is of type I if and only if so is M.Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 38, No. 4, pp. 79–81, 2004Original Russian Text Copyright © by Sh. A. Ayupov     

Generalized Multiplicative Inequalities for Ideal Spaces

We study the problem of completely describing the domains that enjoy the generalized multiplicative inequalities of the embedding theorem type. We transfer the assertions for the Sobolev spaces L _p ¹() to the function classes that result from the replacement of L _p () with an ideal space of vector-functions. We prove equivalence of the functional and geometric inequalities between the norms of indicators and the capacities of closed subsets of . The most comprehensible results relate to the case of the rearrangement invariant ideal spaces.     

The maximal Fejér operator on real Hardy spaces

We prove that the maximal Fej'er operator is not bounded on the real Hardy spaces H ¹, which may be considered over and . We also draw corollaries for the corresponding Hardy spaces over ² and ². This revised version was published online in August 2006 with corrections to the Cover Date.     

Trees, parking functions, syzygies, and deformations of monomial ideals

For a graph , we construct two algebras whose dimensions are both equal to the number of spanning trees of . One of these algebras is the quotient of the polynomial ring modulo certain monomial ideal, while the other is the quotient of the polynomial ring modulo certain powers of linear forms. We describe the set of monomials that forms a linear basis in each of these two algebras. The basis elements correspond to -parking functions that naturally came up in the abelian sandpile model. These ideals are instances of the general class of monotone monomial ideals and their deformations. We show that the Hilbert series of a monotone monomial ideal is always bounded by the Hilbert series of its deformation. Then we define an even more general class of monomial ideals associated with posets and construct free resolutions for these ideals. In some cases these resolutions coincide with Scarf resolutions. We prove several formulas for Hilbert series of monotone monomial ideals and investigate when they are equal to Hilbert series of deformations. In the appendix we discuss the abelian sandpile model.

     

Generalized Sasaki Projections and Riesz Ideals in Pseudoeffect Algebras

A noncommutative version of generalized Sasaki projections in pseudoeffect algebras is introduced. It is proved that an ideal in a pseudoeffect algebra is Riesz if and only if it is closed under the right and left Sasaki projections. In lattice ordered pseudoeffect algebras, it is shown that generalized Sasaki projections are one-element sets, and their explicit form is found. It is shown that if a supremum of a normal Riesz ideal in a lattice ordered pseudoeffect algebra exists, it is a central element. These results extend those obtained recently by Avallone and Vitolo for effect algebras.     

On a Problem of Karpilovsky

Let G be a finite elementary group. Let ⁿ (G) denote the nth power of the augmentation ideal (G) of the integral group ring G. In this paper, we give an explicit basis of the quotient group Q_n(G) = ⁿ(G)/ⁿ⁺¹ (G) and compute the order of Qⁿ (G).2000 Mathematics Subject Classification: 16S34, 20C05     

Prime Rings with Involution Equipped with Some New Product

Let R be a ring with involution *. We consider R as a ring equipped with a new product r s = rs + sr^*. The relationship between (ordinary) ideals of R and right ideals of R with respect to the product is studied.AMS Subject Classification (2000): 16W10, 16D25     

The Lattice of Deductive Systems on Hilbert Algebras

The concept of deductive system on a Hilbert algebra was introduced by A. Diego. We show that the set Ded A of all deductive systems on a Hilbert algebra A forms an algebraic lattice which is distributive.AMS Classification (2000): 06F35, 03G25, 08A30     

关于拟赋值环的一个刻画

Let R be a commutative ring without nil-factor. In this paper, we discuss the problem of quasi-valuation ring presented in the reference “Wang Shianghaw, On quasi-valuation ring, Northeast People‘s Univ. Natur. Sci. J., (1)(1957), 27-40”,when the quotient field of R is an algebraic number field or an algebraic function field, and we obtain a characterization of quasi-valuation rings.