Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1511.02678

On some conditions on a Noetherian ring

C L Wangneo
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper for a noetherian ring R with nilradical N we define semiprime ideals P and Q called as the left and right krull homogenous parts of N . We also recall the known definitions of localisability and the weak ideal invariance (w.i.i for short ) of an ideal of a noetherian ring R . We then state and prove results that culminate in our main theorem whose statement is given below ; Theorem :- Let R be a noetherian ring with nilradical N . Let P and Q be semiprime ideals of R that are the right and left krull homogenous parts of N respectively . Then the following conditions are equivalent ; (i) N is a right w.i.i ideal of R ( respectively N is a left w.i.i ideal of R ) . (ii)P is a right localizable ideal of R ( respectively Q is a left localizable ideal of R ) .

Explore related subjects

math.RA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

C L Wangneo. 2016-04-02. On some conditions on a Noetherian ring. https://arxiv.org/abs/1511.02678

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Maximal tails,character fibres and induced modules for pullback Kumjian-Pask algebras

Let $f:\N^{k}\to\N^{\ell}$ be a surjective monoid homomorphism and let $Γ$ be a row-finite $\ell$-graph with no sources and finitely many vertices. We give an explicit graded isomorphism from the Kumjian--Pask algebra of the pullback $f^{*}Γ$ onto the tensor product of $\KP_{\K}(Γ)$ and the group algebra of the kernel of the group completion of $f$. When $Γ$ is strongly aperiodic, but need not be cofinal, every maximal tail $T$ and every maximal ideal $\mathfrak m$ of the kernel group algebra determine an explicit primitive ideal and primitive quotient. If, in addition, $\K$ is uncountable and algebraically closed, these ideals exhaust the primitive spectrum. We prove that the resulting parametrisation is a homeomorphism for the product of the maximal-tail and Zariski topologies. Each primitive ideal is realised as the annihilator of a simple module induced from the isotropy of a path which is cofinal in $T$, and the character fibres are algebraic tori. Two examples exhibit, respectively, a single character fibre and the non-Hausdorff gluing of two such fibres.

math.RA↗
arXiv2026

A classification of group gradings on incidence algebras over commutative rings

Let $R$ be a commutative ring with 1, $P$ a locally finite partially ordered set, and $G$ a group. We derive necessary and sufficient conditions for an $R$-algebra isomorphism between the incidence algebra $I(P,R)$ and the group algebra $RG$. Then, for an indecomposable ring $R$, a finite poset $P$ and an arbitrary group $G$, we classify the $G$-gradings of $I(P,R)$ up to graded isomorphism. The classification rests on a complete set of primitive orthogonal homogeneous idempotents. The corner algebras are split group algebras of finite abelian subgroups of $G$, and the off-diagonal Peirce blocks are multiplicity-free sums of bimodules induced from characters of double coset stabilizers. Graded isomorphisms are shown to have a rigid form, and a grading is determined up to graded isomorphism by the poset of idempotents, the corner groups, the types of the atomic bimodules and the structure constants of their multiplication. The data which occur are characterized by polynomial conditions, and over an algebraically closed field of characteristic zero only finitely many graded isomorphism classes share given partial invariants. An example shows that the structure constants cannot be omitted. Some previous results are extended and enhanced, while providing alternative proofs for some known facts.

math.RA↗