On the normality of Rees algebras associated to totally unimodular matrices

Let A be an integral matrix with non negative entries and let K be a field. We give a sufficient condition for the normality of the monomial subring determined by the columns of A over the field K. One of the main results proves that if A is totally unimodular, then the Rees algebra of the monomial ideal over K defined by the columns of A is a normal domain.     

Extended Rees algebras and mixed multiplicities

Partially supported by the general research fund at the University of Kansas     

Multi-symbolic Rees algebras and strong F-regularity

Let I be a divisorial ideal of a strongly F-regular ring A. K.-i. Watanabe raised the question whether the symbolic Rees algebra is Cohen-Macaulay whenever it is Noetherian. We develop the notion of multi-symbolic Rees algebras and use this to show that is indeed Cohen-Macaulay whenever a certain auxiliary ring is finitely generated over A. Received August 10, 1998 / in final form October 18, 1999 / Published online July 20, 2000     

Rees algebras with minimal multiplicity

Let (R,m) be a local ring. Let S_M denote the Rees algebra S=R[m^rt] localized at its unique maximal homogeneous ideal M=(m,m^rt). Let TN denote the extended Rees algebra T= R[m^rt, t^-1] localized at its unique maximal homogeneous idea N= (t^?1,m,m^r). Multiplicity formulas are developedfor S_M and T_N. These are used to find necessaIy and sufficient conditions on a Cohen-Macaulay local ring (R,m) and r so that S_M and T_N are Cohen-Macaulay with minimal multiplicity     

On ideal extensions of partial monounary algebras

In the present paper we introduce the notion of an ideal of a partial monounary algebra. Further, for an ideal (I, f _I ) of a partial monounary algebra (A, f _A ) we define the quotient partial monounary algebra (A, f _A )/(I, f _I ). Let (X, f _X ), (Y, f _Y ) be partial monounary algebras. We describe all partial monounary algebras (P, f _P ) such that (X, f _X ) is an ideal of (P, f _P ) and (P, f _P )/(X, f _X ) is isomorphic to (Y, f _Y ). This work was supported by the Slovak VEGA Grant No. 1/3003/06 and by the Science and Technology Assistance Agency under the contract No. APVT-20-004104.     

Coefficient ideals and the Cohen-Macaulay property of Rees algebras

Let be a local ring and let be an ideal of positive height. If is a reduction of , then the coefficient ideal is by definition the largest ideal such that . In this article we study the ideal when the Rees algebra is Cohen-Macaulay.

     

On ideal and subalgebra coefficients in semigroup algebras

Let k[S] be a semigroup algebra with coefficients in a commutative field k, and let U be a one-sided ideal in k[S] or a k-subalgebra of k[S], It is proven that there exists a smallest subfield k′ ≤ k such that U as a one-sided ideal resp. as a k-algebra can be generated by elements in k′[S]. By means of an example it is shown that the straightforward extension of this result to finitely generated commutative k-algebras is not valid.     

Equations defining the multi-Rees algebras of powers of an ideal

In this paper we describe the defining equations of the multi-Rees algebra $R[{\{I^{n_i}{t}_i\}}_{1\le i\le r}]$ over a Noetherian ring R when I is an ideal of linear type. This generalizes a result of Ribbe and recent work of Lin–Polini and Sosa.     

Weighted Rees matrix semigroup algebras and their applications

In this work, we will describe the weighted semigroup algebra ? ¹(S, ω), where S is a regular Rees matrix semigroup and ω ≥  1. Then as an application, we investigate the amenability of the semigroup algebra ? ¹(S, ω) and its second dual for an arbitrary semigroup S.     

On the cofinalities of Boolean algebras and the ideal of null sets

We will show that if the cofinality of the ideal of Lebesgue measure zero sets is equal to then there exists a Boolean algebra B of cardinality which is not a union of strictly increasing -sequence of its subalgebras. This generalizes a result of Just and Koszmider who showed that it is consistent with that such an algebra exists. Received November 21; accepted in final form April 12, 2001.     

Regular sequences in Rees and symmetric algebras I

In this paper we describe necessary and sufficient conditions for a system of elements a₁,...,a_t of a local Noetherian ring A such that the sequence a₁T,a₁–a₂T,...,a_t–1– a_tT, a_tin the Rees algebra A[a₁T,...,a_tT], T is an indeterminate, constitutes a regular sequence.     

Regular sequences in Rees and symmetric algebras,II

In continuing [7] we study necessary and sufficient conditions for a system of elements b₁,...,b_s,a₁,...,a_t of a local Noetherian ring A such that the sequence b₁,...,b_s,a₁T,a₁-a₂T,...,a_t–1-a_tT,a_t in the Rees algebra A[a₁ T,...,a_t T], T is an indeterminate, constitutes a regular sequence.     

Homomorphisms of $$\ell ^1$$-Munn algebras and Rees matrix semigroup algebras

Let \({\mathcal {LM}}\left( {\mathcal {A}}, P\right) \) be an \(\ell ^1\)-Munn algebra over an arbitrary unital Banach algebra \({\mathcal {A}}\). We characterize homomorphisms from \({\mathcal {LM}}\left( {\mathcal {A}}, P\right) \) into an arbitrary Banach algebra \({\mathcal {B}}\) in terms of homomorphisms from \({\mathcal {A}}\) into \({\mathcal {B}}\). Then we discuss homomorphisms from arbitrary Banach algebras into \({\mathcal {LM}}\left( {\mathcal {A}}, P\right) \). Existence and uniqueness of homomorphisms under certain conditions are also discussed. We apply these results to the concrete case of \(\ell ^1(S)\) where S is a Rees matrix semigroup, to identify characters of \(\ell ^1(S)\) in both cases where S is with or without zero. As a consequence if the sandwich matrix of S has a zero entry, then \(\ell ^1(S)\) is character amenable.