The adcending chain condition for real ideas.
The containment property for large quotient rings.
The Divisor Class Group of Ordinary and Symbolic Blow-ups.
The effect on associated prime ideals produced by an extension of the base field.
The First Isomorphism Theorem and Other Properties of Rings
Different properties of rings and fields are discussed [12], [41] and [17]. We introduce ring homomorphisms, their kernels and images, and prove the First Isomorphism Theorem, namely that for a homomorphism f : R → S we have R/ker(f) ≅ Im(f). Then we define prime and irreducible elements and show that every principal ideal domain is factorial. Finally we show that polynomial rings over fields are Euclidean and hence also factorial
The GCD property and irreducible quadratic polynomials.
The Heigth of Ideals and Regular Sequences.
The lattice of radical filters of a commutative Noetherian ring
The least solution for the polynomial interpolation problem.
The Lifting of determinal prime ideals.
The maximal regular ideal of some commutative rings
In 1950 in volume 1 of Proc. Amer. Math. Soc., B. Brown and N. McCoy showed that every (not necessarily commutative) ring has an ideal consisting of elements for which there is an such that , and maximal with respect to this property. Considering only the case when is commutative and has an identity element, it is often not easy to determine when is not just the zero ideal. We determine when this happens in a number of cases: Namely when at least one of or has a von Neumann inverse,...
The Ohm type properties for multiplication ideals.
The Primary components of and Integral Closures of Ideals in 3-Dimensional Regular Local Rings.
The prime ideals intersection graph of a ring
Let be a commutative ring with unity and be the set of unit elements of . In this paper, we introduce and investigate some properties of a new kind of graph on the ring , namely, the prime ideals intersection graph of , denoted by . The is a graph with vertex set and two distinct vertices and are adjacent if and only if there exists a prime ideal of such that . We obtain necessary and sufficient conditions on such that is disconnected. We find the diameter and girth of ....
The projective spectrum as a distributive lattice
The -ideals in polynomial rings and the -modules over polynomial rings.
The -transform and its dual with applications to Prüfer extensions
The Structure of Certain Ideal Transforms.
The theta ideal, dense submodules and the forcing linearity number for a multiplication module.
The total torsion element graph of a module over a commutative ring.