On 2r-ideals in commutative rings with zero-divisors

Let R be a commutative ring with nonzero identity. Let 𝒥( R ) be the set of all ideals of R and let δ : 𝒥 ( R ) → 𝒥 ( R ) be a function. Then δ is called an expansion function of ideals of R if whenever L, I, J are ideals of R with J ⊆ I , we have L ⊆ δ ( L ) and δ ( J ) ⊆ δ ( I ). Let δ be an expansion function of ideals of R . In this paper, we introduce and investigate a new class of ideals that is closely related to the class of δ -primary ideals. A proper ideal I of R is said to be a 1-absorbing δ -primary ideal if whenever nonunit elements a, b, c ∈ R and abc ∈ I , then ab ∈ I or c ∈ δ ( I ). Moreover, we give some basic properties of this class of ideals and we study the 1-absorbing δ -primary ideals of the localization of rings, the direct product of rings and the trivial ring extensions.

In this chapter, we characterize prime fuzzy ideals of a semigroup S. Sections 7.1–7.11 are essentially from [151]. We show that a nonconstant fuzzy ideal f of a semigroup S is prime if and only if f is two-valued and there exists an element x0 in S such that f(x0) = 1 and f1 = {x ∈ S | f(x) = 1} is a prime ideal of S. We also consider the concepts of weakly prime fuzzy ideals, completely prime fuzzy ideals and weakly completely prime fuzzy ideals of a semigroup S. We establish relations among four classes of prime ideals.

This series aims to report new developments in mathematical research and teaching -quickly, informally and at a high level.The type of material considered for publication includes: 1.Preliminary drafts of original papers and monographs 2. Lectures on a new field, or presenting a new angle on a classical field 3. Seminar work-outs 4. Reports of meetings, provided they are a) of exceptional interest or b) devoted to a single topic.

Integral domains are commutative rings whose non-zero elements are closed under multiplication. If each nonzero element is a unit, the domain is called a field and is shipped off to Chap. 11. For the domains D which remain, divisibility is a central question. A prime ideal has the property that elements outside the ideal are closed under multiplication. A non-zero element \(a\in D\) is said to be prime if the principle ideal Da which it generates is a prime ideal. D is a unique factorization domain (or UFD) if any expression of an element as a product of prime elements is unique up to the order of the factors and the replacement of any prime factor by a unit multiple. If D is a UFD, so is the polynomial ring D[X] where X is a finite set of commuting indeterminates. In some cases, the unique factorization property can be determined by the localizations of a domain. Euclidean domains (like the integers, Gaussian and Eisenstein numbers) are UFD’s, but many domains are not. One enormous class of domains (which includes the algebraic integers) is obtained the following way: Suppose K a field which is finite-dimensional over a subfield F which, in turn, is the field of fractions of an integral domain D. One can then define the ring \(\mathcal{O}_D(K)\) of elements of K which are integral with respect to D. Under modest conditions, the integral domain \(\mathcal{O}_D(K)\), will become a Noetherian domain in which every prime ideal is maximal—a so-called Dedekind domain. Although not UFD’s, Dedekind domains offer a door prize: every ideal can be uniquely expressed as a product of prime ideals (up to the order of the factors, of course).

Let K be a field and K[x] the polynomial ring over K in an indeterminate x. If P is a prime ideal of K[x], then there exists an irreducible polynomial f in K[x] such that P = K[x]f. This result is quite old and basic; however no corresponding result seems to be known for a polynomial ring in n indeterminates x1, ..., xn over K. Actually, it seems to be very difficult to find some system of generators for a prime ideal of K[xl,...,Xn]. Now, K [x1, ..., xn] is a Noetherian ring and by a converse of the principal ideal theorem for every prime ideal P of K[x1, ..., xn] there exist n polynomials fi, f..n such that P is minimal over (fi, ..., fn), the ideal generated by { fl, ..., f}n ([4], Theorem 153). Also, as a consequence of ([1], Theorem 1) it follows that any prime ideal of K[xl, X.., Xn] is determined by n polynomials. However it is not clear in [1] how to find these polynomials and no converse result is proved. The purpose of this paper is to show a result which in particular implies that every prime ideal of K[xi, ..., xn] is determined by a sequence of n polynomials which is in some sense irreducible, and such that the converse is also true. In general, let R be any prime ring. If P is a nonzero prime ideal of the polynomial ring R[x] with P n R = 0 (an R-disjoint prime ideal), then P = Q[x]fo n R[x] where Q is a ring of right quotients of R and fo E C[x] is an irreducible polynomial, C being the extended centroid of R ([2], Corollary 2.7). This characterization has the problem that we have to compute Q to have a prime ideal determined. A better way (an intrinsic one) to determine an R-disjoint prime ideal has been given in [3]. We proved that every R-disjoint prime ideal P of R[x] is determined by just one polynomial in P which is in some sense irreducible ([3], Theorem 1.4).

Almost complete intersections and factorial rings

Let R be a commutative ring with a nonzero unity element. In this article, we introduce and examine new classes of ideals in Z2-graded rings, expanding upon the earlier concept of N-prime ideals. Using the function N : R → R0, defined by N(x) = x0 2 – x1 2 for x = x0 + x1 ∈ R, we define and analyze two new types of ideals, 2-N-prime and weakly 2-N-prime ideals. A proper ideal I is called 2-N-prime if xy ∈ I implies that N(x2) ∈ I or N(y2) ∈ I. Similarly, I is weakly 2-N-prime if 0 ≠ xy ∈ I implies that N(x2) ∈ I or N(y2) ∈ I. We explore the fundamental properties of these ideals, highlighting their connections to established structures in graded ring theory. Through examples and counter examples, we illustrate the differences between 2-N-prime and weakly 2-N-prime ideals, underscoring their significance in the theory of graded rings. These results not only deepensour understanding of ideal theory in Z2-graded rings but also open new avenues for futures research.

Abstract. Let R be a ring and I be a proper ideal of R . For the caseof R being commutative, Anderson proved that ( ∗ ) there are only finitelymany prime ideals minimal over I whenever every prime ideal minimalover I is finitely generated. We in this note extend the class of ringsthat satisfies the condition ( ∗ ) to noncommutative rings, so called ho-momorphically IFP , which is a generalization of commutative rings. Asa corollary we obtain that there are only finitely many minimal primeideals in the polynomial ring over R when every minimal prime ideal ofa homomorphically IFP ring R is finitely generated. Throughout every ring is associative with identity unless otherwise stated.The n by n matrix ring over a ring R is denoted by Mat n ( R ). Due to Bell [2], aright (or left) ideal I of a ring R is said to have the insertion-of-factors-property (simply IFP ) if ab ∈ I implies aRb ⊆ I for a,b ∈ R . A ring R is called IFP ifthe zero ideal of R has the IFP. For a ring R and an ideal I , note that

Let [Formula: see text] be a field and [Formula: see text] be a polynomial ring in the variables [Formula: see text]. In this paper, we introduce two classes of monomial ideals of [Formula: see text], which have the following properties: (i) The (strong) persistence property of associated prime ideals. (ii) There exists a strongly superficial element. (iii) Ratliff–Rush closed. Next, we characterize these monomial ideals. In the sequel, we give some combinatorial aspects. We conclude this paper with constructing new monomial ideals, which have the persistence property.

Among commutative rings, the polynomial rings in a finite number of indeterminates enjoy important special properties and are frequently used in applications. As they are also of paramount importance in Algebraic Geometry, polynomial rings have been intensively studied. On the other hand, rings of formal power series have been extensively used in “algebroid geometry” and have many properties which are parallel to those of polynomial rings. In the first section of this chapter we shall define formal power series rings and we shall show that the main properties of polynomial rings which have been derived in previous chapters (see, in particular, Vol. I, Ch. I, §§ 16–18) hold also for formal power series rings. In the later sections of this chapter we shall give deeper properties of polynomial rings and, whenever possible, the parallel properties of power series rings.

Let R be a commutative ring with identity and $${\mathbb {N}}_0$$ be the additive monoid of nonnegative integers. We say that a function $$t : {\mathbb {N}}_0 \times {\mathbb {N}}_0 \rightarrow R$$ is a twist function on R if t satisfies the following three properties for all $$n, m, q \in {\mathbb {N}}_0$$ : (i) $$t(0,q) = 1$$ , (ii) $$t(n,m) = t(m,n)$$ , and (iii) $$t(n,m) \cdot t(n + m, q) = t (n, m + q) \cdot t(m, q)$$ . Let $$R[\![X]\!]$$ (resp., R[X]) be the set of power series (resp., polynomials) with coefficients in R. For $$f = \sum _{n=0}^{\infty } a_nX^n$$ and $$g = \sum _{n=0}^{\infty } b_nX^n \in R[\![X]\!]$$ , let $$f+g = \sum _{n=0}^{\infty } (a_n+b_n)X^n$$ , $$f*_tg = \sum _{n=0}^{\infty }(\sum _{i+j = n}t(i,j)a_ib_j)X^n$$ . Then, $$R^t[\![X]\!]:= (R[\![X]\!], +, *_t)$$ and $$R^t[X] := (R[X], +, *_t)$$ are commutative rings with identity that contain R as a subring. In this paper, we study ring-theoretic properties of $$R^t[\![X]\!]$$ and $$R^t[X]$$ with focus on divisibility properties including UFDs and GCD-domains. We also show how these two rings are related to the usual power series and polynomial rings.

Prime ideals in ultraproducts of commutative rings

In this paper, we introduce $$\phi $$ -1-absorbing prime ideals in commutative rings. Let R be a commutative ring with a nonzero identity $$1\ne 0$$ and $$\phi :\mathcal {I}(R)\rightarrow \mathcal {I}(R)\cup \{\emptyset \}$$ be a function where $$\mathcal {I}(R)$$ is the set of all ideals of R. A proper ideal I of R is called a $$\phi $$ -1-absorbing prime ideal if for each nonunits $$x,y,z\in R$$ with $$xyz\in I-\phi (I)$$ , then either $$xy\in I$$ or $$z\in I$$ . In addition to give many properties and characterizations of $$\phi $$ -1-absorbing prime ideals, we also determine rings in which every proper ideal is $$\phi $$ -1-absorbing prime.

Knutson ideals and determinantal ideals of Hankel matrices

Let R be a commutative ring with identity and S be a multiplicative subset of R . In this paper we introduce the concept of almost S-prime ideal as a new generalization of S−prime ideal. Let P be a proper ideal of R disjoint with S. Then P is said to be almost S- prime ideal if there exists s ∈ S such that, for all x, y ∈ R if xy ∈ P − P 2 then sx ∈ P or sy ∈ P. Number of results concerning this concept and examples are given. Furthermore, we investigate an almost S- prime ideals of trivial ring extensions and amalgamation rings..