Let A be the ring obtained by localizing the polynomial ring κ[X, Y, Z, W ] over a field κ at the maximal ideal (X, Y, Z, W) and modulo the ideal (XW − Y Z). Let p be the ideal of A generated by X and Y. We study the module structure of a minimal injective resolution of A/p in details using local cohomology. Applications include the description of Ext i A (M, A/p), where M is a module construct...

Let A be the ring obtained by localizing the polynomial ring κ[X,Y, Z,W ] over a field κ at the maximal ideal (X, Y, Z,W ) and modulo the ideal (XW − Y Z). Let p be the ideal of A generated by X and Y . We study the module structure of a minimal injective resolution of A/p in details using local cohomology. Applications include the description of Ext A (M,A/p), where M is a module constructed b...

We investigate perturbative thermodynamic geometry of nonextensive ideal classical, Bose, and Fermi gases. We show that the intrinsic statistical interaction of nonextensive Bose (Fermi) gas is attractive (repulsive) similar to the extensive case but the value of thermodynamic curvature is changed by a nonextensive parameter. In contrary to the extensive ideal classical gas, the nonextensive on...

Given two Banach spaces X and Y , we write L(X, Y ) for the space of all continuous linear operators from X to Y . A linear subspace J of L(X, Y ) is said to be an ideal if ATB ∈ J whenever A ∈ L(Y ), T ∈ J , and B ∈ L(X). It is known (see, e.g., Caradus:74 [CPY74]) that the only norm closed ideal in L(lp), 1 6 p < ∞ is the ideal of compact operators. The structure of closed ideals in L(lp ⊕ lq...

We construct a Banach space Z such that the lattice of closed two-sided ideals of the Banach algebra B(Z) of bounded operators on Z is as follows: {0} ⊂ K (Z) ⊂ E (Z) ⊂ ⊂ M1 M2 ⊂ ⊂ We then determine which kinds of approximate identities (bounded/left/right), if any, each of the four non-trivial closed ideals of B(Z) contains, and we show that the maximal ideal M1 is generated as a left ideal by...

the minimal prime decomposition for semiprime ideals is defined and studied on z-ideals of c(x). the necessary and sufficient condition for existence of the minimal prime decomposition of a z-ideal / is given, when / satisfies one of the following conditions: (i) / is an intersection of maximal ideals. (ii) i is an intersection of o , s, when x is basically disconnected. (iii) i=o , when x x ha...

The minimal prime decomposition for semiprime ideals is defined and studied on z-ideals of C(X). The necessary and sufficient condition for existence of the minimal prime decomposition of a z-ideal / is given, when / satisfies one of the following conditions: (i) / is an intersection of maximal ideals. (ii) I is an intersection of O , s, when X is basically disconnected. (iii) I=O , when x X h...

‎In this paper‎, ‎we define and study the notion of zero elements in topoframes; a topoframe is a pair‎ ‎$(L‎, ‎tau)$‎, ‎abbreviated $L_{ tau}$‎, ‎consisting of a frame $L$ and a‎ ‎subframe $ tau $ all of whose elements are complemented elements in‎ ‎$L$‎. ‎We show that‎ ‎the $f$-ring $ mathcal{R}(L_tau)$‎, ‎the set of $tau$-real continuous functions on $L$‎, ‎is uniformly complete‎. ‎Also‎, ‎t...

In this paper we explore the interrelationship between the theory of polynomial ideals and certain branches of analysis including multivariate approximation theory and linear partial differential equations. Specifically, we use the perturbation technique from analysis to study the codimension of a multivariate polynomial ideal. Let K be a field. In this paper, K will often be the field C of com...

We say that a Tychonoff space X has computable z-radicals if for all ideals a of C(X), the smallest z-ideal containing a is generated as an ideal by all the s◦f , where f is in a and s is a continuous function IR −→ IR with s−1(0) = {0}. We show that every cozero set of a compact space has computable z-radicals and that a subset X of IR has computable z-radicals if and only if X is locally closed.