4 research outputs found
Ratliff–Rush filtrations associated with ideals and modules over a Noetherian ring
No full textAbstractLet R be a commutative Noetherian ring, M a finitely generated R-module and I a proper ideal of R. In this paper we introduce and analyze some properties of r(I,M)=⋃k⩾1(Ik+1M:IkM), the Ratliff–Rush ideal associated with I and M. When M=R (or more generally when M is projective) then r(I,M)=I˜, the usual Ratliff–Rush ideal associated with I. If I is a regular ideal and annM=0 we show that {r(In,M)}n⩾0 is a stable I-filtration. If Mp is free for all p∈SpecR∖m-SpecR, then under mild condition on R we show that for a regular ideal I, ℓ(r(I,M)/I˜) is finite. Further r(I,M)=I˜ if A∗(I)∩m-SpecR=∅ (here A∗(I) is the stable value of the sequence Ass(R/In)). Our generalization also helps to better understand the usual Ratliff–Rush filtration. When I is a regular m-primary ideal our techniques yield an easily computable bound for k such that In˜=(In+k:Ik) for all n⩾1. For any ideal I we show that InM˜=InM+HI0(M) for all n≫0. This yields that R˜(I,M)=⊕n⩾0InM˜ is Noetherian if and only if depthM>0. Surprisingly if dimM=1 then G˜I(M)=⊕n⩾0InM˜/In+1M˜ is always a Noetherian and a Cohen–Macaulay GI(R)-module. Application to Hilbert coefficients is also discussed
Matrix representations of linear transformations on bicomplex space
Full text linkAn algebraic investigation on bicomplex numbers is carried out here.
Particularly matrices and linear maps defined on them are discussed. A new kind
of cartesian product, referred to as an idempotent product, is introduced and
studied. The elements of this space are linear maps of a special form. These
linear maps are examined with respect to usual notions like kernel, range, and
singularity. Their matrix representation is also discussed
A Generalization to Ordinary Derivative and its Associated Integral with some applications
No full textThis paper proposes a generalization to the ordinary derivative, the deformable derivative. For this, we employ a limit approach like theordinary derivative but use a parameter varying over the unit interval. Thedefinition makes the deformable derivative equivalent to the ordinary derivativebecause one’s existence implies another. Its intrinsic property ofcontinuously deforming function to its derivative, together with the graphicalillustration of linear expression of the function and its derivative, renderssufficient substances to name it deformable derivative. We deriveRolle’s, Mean-value and Taylor’s theorems for the deformable derivativeby establishing some of its basic properties. We also define the deformableintegral using the fundamental theorem of calculus and discuss associatedinverse, linearity, and commutativity property. In addition, we establish aconnection between deformable integral and Riemann-Liouville fractionalintegral. As theoretical applications, we solve some fractional differentialequation
