Related papers: Prufer conditions in an amalgamated duplication of…
We prove an equivalent condition for the existence of a link between prime ideals in terms of the structure of a certain cohomology module. We use this formulation to answer an open question regarding the nature of module extensions over…
We develop the theory of Gr\"obner bases for ideals in a polynomial ring with countably infinite variables over a field. As an application we reconstruct some of the one-one correspondences among various sets of partitions by using division…
The theory of generalized inverses of matrices and operators is closely connected with projections, i.e., idempotent (bounded) linear transformations. We show that a similar situation occurs in any associative ring $\mathcal{R}$ with a unit…
In this paper, we investigate the algebraic properties of the expectation semirings which are semiring version of the concept of trivial extension in ring theory. We discuss ideals, primes, maximals and primary ideals of these semirings. We…
We develop a new technique for studying monomial ideals in the standard polynomial rings $A[X_1,\ldots,X_d]$ where $A$ is a commutative ring with identity. The main idea is to consider induced ideals in the semigroup ring…
This paper mainly focuses on commutative local domains of dimension one. We then obtain a criterion for a ring to have a finite number of trace ideals in terms of integrally closed ideals. We also explore properties of such rings related to…
In the theory of commutative semirings, the lack of additive inverses creates a structural divergence between ideals and congruences that does not exist in ring theory. The aim of this article is to restore critical ideal-theoretic…
In this paper, the structure of the ideals in the ring of Colombeau generalized numbers is investigated. Connections with the theories of exchange rings, Gelfand rings and lattice-ordered rings are given. Characterizations for prime,…
The aim of this work is to study duality of fractional ideals with respect to a fixed ideal and to investigate the relationship between value sets of pairs of dual ideals in admissible rings, a class of rings that contains the local rings…
In this paper, we introduced the concept of a $p$-ideal for a given ring. We provide necessary and sufficient condition for $\dfrac{R[x]}{(f(x))}$ to be a $p$-ring, where $R$ is a finite $p$-ring. It is also shown that the amalgamation of…
For a commutative ring R we investigate the property that the sets of minimal primes of finitely generated ideals of R is always finite. We prove this property passes to polynomial ring extensions (in an arbitrary number of variables) over…
We estimate the Castelnuovo-Mumford regularity of ideals in a polynomial ring over a field by studying the regularity of certain modules generated in degree zero and with linear relations. In dimension one, this process gives a new type of…
We prove some algebraic results on the ring of matrix differential operators over a differential field in the generality of non-commutative principal ideal rings. These results are used in the theory of non-local Poisson structures.
We generalize signature Gr\"obner bases, previously studied in the free algebra over a field or polynomial rings over a ring, to ideals in the mixed algebra $R[x_1,...,x_k]\langle y_1,\dots,y_n \rangle$ where $R$ is a principal ideal…
Given a unital associative ring S and a subring R, we say that S is an ideal (or Dorroh) extension of R if for some ideal I of S, S = R + I, where the sum is direct. In this note we investigate the ideal structure of an arbitrary ideal…
In this work, we extend the definition of the graded prime ideals from those in commutative graded rings to the ideals over graded Lie algebras. We prove some facts about graded prime Lie ideals in arbitrary Lie algebras that are similar to…
In this expositional paper, we discuss commutative algebra -- a study inspired by the properties of integers, rational numbers, and real numbers. In particular, we investigate rings and ideals, and their various properties. After, we…
A commutative ring is reduced when it can be embedded into a direct product of fields. While the category of reduced commutative rings plays a fundamental role in affine geometry, it exhibits several structural deficiencies: it admits…
Given a polynomial ring $C$ over a field and proper ideals $I$ and $J$ whose generating sets involve disjoint variables, we determine how to embed the associated primes of each power of $I+J$ into a collection of primes described in terms…
This paper is a continuation of a previous work by the author and G. Puninski where iterated intersections of powers of ideals were studied in rings of iterated differential polynomials. We present a method which can be used to show that…