This paper investigates situations where a property of a ring can be tested on a set of “prime right ideals.” Generalizing theorems of Cohen and Kaplansky, we show that every right ideal of a ring is finitely generated (resp. principal) iff every “prime right ideal” is finitely generated (resp. principal), where the phrase “prime right ideal” can be interpreted in one of many different ways. We...

Let [Formula: see text] be a commutative ring and the multiplicative group of unit elements text]. In 2012, Khashyarmanesh et al. defined generalized unitary Cayley graph, text], corresponding to subgroup nonempty subset with as graph vertex set text]and two distinct vertices being adjacent if only there exists such that this paper, we characterize all Artinian rings for which is projective. Th...

Let A⊆B be a ring extension and G set of A-submodules B. We introduce class closure operations on (which we call multiplicative (A,B,G)) that generalizes the classes star, semistar semiprime operations. study how Mult(A,B,G) these varies when A, B or vary, behaves under homomorphisms. As an application, show to reduce star analytically unramified one-dimensional Noetherian domains closures fini...

A local ring is an associative with unique maximal ideal. We associate each Artinian singleton basis four invariants (positive integers) p,n,s,t. The purpose of this article to describe the structure such rings and classify them (up isomorphism) same invariants. Every can be constructed over its coefficient subring by a certain polynomial called associated polynomial. These polynomials play sig...

Let [Formula: see text] be an text]-finite, Krull–Schmidt and text]-linear text]-exangulated category with a commutative artinian ring. In this note, we prove that has Auslander–Reiten–Serre duality if only Auslander–Reiten text]-exangles. Moreover, also give equivalent condition for the existence of Serre (which is special type duality). Finally, assume further duality. We exploit bijection tr...

Let R be a commutative ring and G(R) be a graph with vertices as proper andnon-trivial ideals of R. Two distinct vertices I and J are said to be adjacentif and only if I + J = R. In this paper we study a graph constructed froma subgraph G(R)Δ(R) of G(R) which consists of all ideals I of R such thatI Δ J(R), where J(R) denotes the Jacobson radical of R. In this paper westudy about the relation b...

It is proved that a commutative ring is clean if and only if it is Gelfand with a totally disconnected maximal spectrum. It is shown that each indecomposable module over a commutative ring R satisfies a finite condition if and only if R P is an artinian valuation ring for each maximal prime ideal P. Commutative rings for which each indecomposable module has a local endomorphism ring are studied...

Let R be a polynomial ring in r variables over an infinite field K, and denote by D a corresponding dual ring, upon which R acts as differential operators. We study type two graded level Artinian algebras A = R/I , having socle degree j. For each such algebra A, we consider the family of Artinian Gorenstein [AG] quotients of A having the same socle degree. By Macaulay duality, A corresponds to ...

Let R be a commutative ring and Max (R) be the set of maximal ideals of R. The regular digraph of ideals of R, denoted by −−→ Γreg(R), is a digraph whose vertex set is the set of all non-trivial ideals of R and for every two distinct vertices I and J , there is an arc from I to J whenever I contains a J-regular element. The undirected regular (simple) graph of ideals of R, denoted by Γreg(R), h...