scholarly journals PROJECTIVE PRIME IDEALS AND LOCALISATION IN PI-RINGS

2001 ◽  
Vol 64 (1) ◽  
pp. 1-12 ◽  
Author(s):  
A. W. CHATTERS ◽  
C. R. HAJARNAVIS ◽  
R. M. LISSAMAN
Keyword(s):  
Prime Ideal ◽  
Maximal Ideal ◽  
Regular Element ◽  
Prime Ideals ◽  
Ideal Ring ◽  
Left And Right

The results here generalise [2, Proposition 4.3] and [9, Theorem 5.11]. We shall prove the following.THEOREM A. Let R be a Noetherian PI-ring. Let P be a non-idempotent prime ideal of R such that PRis projective. Then P is left localisable and RPis a prime principal left and right ideal ring.We also have the following theorem.THEOREM B. Let R be a Noetherian PI-ring. Let M be a non-idempotent maximal ideal of R such that MRis projective. Then M has the left AR-property and M contains a right regular element of R.

Minimal prime ideals and arithmetic comprehension

1991 ◽  
Vol 56 (1) ◽  
pp. 67-70 ◽  
Author(s):  
Kostas Hatzikiriakou
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Maximal Ideal ◽  
Commutative Rings ◽  
Minimal Prime Ideal ◽  
Prime Ideals ◽  
Order Arithmetic ◽  
Minimal Prime ◽  
Weak König’S Lemma ◽  
König’S Lemma

We assume that the reader is familiar with the program of “reverse mathematics” and the development of countable algebra in subsystems of second order arithmetic. The subsystems we are using in this paper are RCA0, WKL0 and ACA0. (The reader who wants to learn about them should study [1].) In [1] it was shown that the statement “Every countable commutative ring has a prime ideal” is equivalent to Weak Konig's Lemma over RCA0, while the statement “Every countable commutative ring has a maximal ideal” is equivalent to Arithmetic Comprehension over RCA0. Our main result in this paper is that the statement “Every countable commutative ring has a minimal prime ideal” is equivalent to Arithmetic Comprehension over RCA0. Minimal prime ideals play an important role in the study of countable commutative rings; see [2, pp. 1–7].

scholarly journals On 2-absorbing ideals of commutative rings

2007 ◽  
Vol 75 (3) ◽  
pp. 417-429 ◽  
Author(s):  
Ayman Badawi
Keyword(s):  
Prime Ideal ◽  
Maximal Ideal ◽  
Integral Domain ◽  
Commutative Rings ◽  
Dedekind Domain ◽  
Proper Ideal ◽  
Prime Ideals ◽  
Maximal Ideals ◽  
Noetherian Domain ◽  
Valuation Domains

Suppose that R is a commutative ring with 1 ≠ 0. In this paper, we introduce the concept of 2-absorbing ideal which is a generalisation of prime ideal. A nonzero proper ideal I of R is called a 2-absorbing ideal of R if whenever a, b, c ∈ R and abc ∈ I, then ab ∈ I or ac ∈ I or bc ∈ I. It is shown that a nonzero proper ideal I of R is a 2-absorbing ideal if and only if whenever I1I2I3 ⊆ I for some ideals I1,I2,I3 of R, then I1I2 ⊆ I or I2I3 ⊆ I or I1I3 ⊆ I. It is shown that if I is a 2-absorbing ideal of R, then either Rad(I) is a prime ideal of R or Rad(I) = P1 ⋂ P2 where P1,P2 are the only distinct prime ideals of R that are minimal over I. Rings with the property that every nonzero proper ideal is a 2-absorbing ideal are characterised. All 2-absorbing ideals of valuation domains and Prüfer domains are completely described. It is shown that a Noetherian domain R is a Dedekind domain if and only if a 2-absorbing ideal of R is either a maximal ideal of R or M2 for some maximal ideal M of R or M1M2 where M1,M2 are some maximal ideals of R. If RM is Noetherian for each maximal ideal M of R, then it is shown that an integral domain R is an almost Dedekind domain if and only if a 2-absorbing ideal of R is either a maximal ideal of R or M2 for some maximal ideal M of R or M1M2 where M1,M2 are some maximal ideals of R.

scholarly journals On 2-absorbing ideals of commutative semirings

2019 ◽  
Vol 19 (02) ◽  
pp. 2050034
Author(s):  
H. Behzadipour ◽  
P. Nasehpour
Keyword(s):  
Prime Ideal ◽  
Maximal Ideal ◽  
Proper Ideal ◽  
Prime Ideals ◽  
Ideal Formula ◽  
Minimal Prime ◽  
Unique Maximal Ideal

In this paper, we investigate 2-absorbing ideals of commutative semirings and prove that if [Formula: see text] is a nonzero proper ideal of a subtractive valuation semiring [Formula: see text] then [Formula: see text] is a 2-absorbing ideal of [Formula: see text] if and only if [Formula: see text] or [Formula: see text] where [Formula: see text] is a prime ideal of [Formula: see text]. We also show that each 2-absorbing ideal of a subtractive semiring [Formula: see text] is prime if and only if the prime ideals of [Formula: see text] are comparable and if [Formula: see text] is a minimal prime over a 2-absorbing ideal [Formula: see text], then [Formula: see text], where [Formula: see text] is the unique maximal ideal of [Formula: see text].

scholarly journals On two-sided artinian quotient rings

1972 ◽  
Vol 13 (2) ◽  
pp. 159-163 ◽  
Author(s):  
P. F. Smith
Keyword(s):  
Prime Ideal ◽  
Noetherian Ring ◽  
Regular Element ◽  
Quotient Ring ◽  
Prime Ideals ◽  
Commutative Noetherian Ring ◽  
Zero Divisors ◽  
Quotient Rings

Djabali [1] has proved that, if R is a right and left noetherian ring with an identity and if the proper prime ideals of R are maximal, then R has a right and left artinian two-sided quotient ring. Robson [5, Theorem 2.11] and Small [6, Theorem 2.13] have proved independently that, if Ris a commutative noetherian ring, then Rhas an artinian quotient ring if and only if the prime ideals of Rthat belong to the zero ideal are all minimal. We shall generalise these results by proving theTheorem. Let R be a right and left noetherian ring with a regular element. Then R has a right and left artinian two-sided quotient ring if and only if each prime ideal of R consisting of zero–divisors is minimal.

scholarly journals One sided invertibility and localisation

1992 ◽  
Vol 34 (3) ◽  
pp. 333-339 ◽  
Author(s):  
C. R. Hajarnavis
Keyword(s):  
Prime Ideal ◽  
Maximal Ideal ◽  
Polynomial Identity ◽  
Noetherian Ring ◽  
Crucial Role ◽  
Prime Ideals ◽  
Careful Study ◽  
Definition Of ◽  
The Right

In general, a prime ideal P of a prime Noetherian ring need not be classically localisable. Since such a localisation, when it does exist, is a striking property; sufficiency criteria which guarantee it are worthy of careful study. One such condition which ensures localisation is when P is an invertible ideal [5, Theorem 1.3]. The known proofs of this result utilise both the left as well as the right invertiblity of P. Such a requirement is, in practice, somewhat restrictive. There are many occasions such as when a product of prime ideals is invertible [6] or when a non-idempotent maximal ideal is known to be projective only on one side [2], when the assumptions lead to invertibilty also on just one side. Our main purpose here is to show that in the context of Noetherian prime polynomial identity rings, this one-sided assumption is enough to ensure classical localisation [Theorem 3.5]. Consequently, if a maximal ideal in such a ring is invertible on one side then it is invertible on both sides [Proposition 4.1]. This result plays a crucial role in [2]. As a further application we show that for polynomial identity rings the definition of a unique factorisation ring is left-right symmetric [Theorem 4.4].

scholarly journals Intuitionistic fuzzy normed prime and maximal ideals

2021 ◽  
Vol 6 (10) ◽  
pp. 10565-10580
Author(s):  
Nour Abed Alhaleem ◽  
◽  
Abd Ghafur Ahmad
Keyword(s):  
Characteristic Function ◽  
Prime Ideal ◽  
Maximal Ideal ◽  
Prime Ideals ◽  
Intuitionistic Fuzzy ◽  
Normed Ideal ◽  
Maximal Ideals

<abstract><p>Motivated by the new notion of intuitionistic fuzzy normed ideal, we present and investigate some associated properties of intuitionistic fuzzy normed ideals. We describe the intrinsic product of any two intuitionistic fuzzy normed subsets and show that the intrinsic product of intuitionistic fuzzy normed ideals is a subset of the intersection of these ideals. We specify the notions of intuitionistic fuzzy normed prime ideal and intuitionistic fuzzy normed maximal ideal, we present the conditions under which a given intuitionistic fuzzy normed ideal is considered to be an intuitionistic fuzzy normed prime (maximal) ideal. In addition, the relation between the intuitionistic characteristic function and prime and maximal ideals is generalized. Finally, we characterize relevant properties of intuitionistic fuzzy normed prime ideals and intuitionistic fuzzy normed maximal ideals.</p></abstract>

scholarly journals On the essentiality and primeness of λ-super socle of C(X)

2018 ◽  
Vol 19 (2) ◽  
pp. 261
Author(s):  
S. Mehran ◽  
M. Namdari ◽  
S. Soltanpour
Keyword(s):  
Prime Ideal ◽  
Maximal Ideal ◽  
Topological Property ◽  
Regular Cardinal ◽  
Cardinal Number ◽  
Prime Ideals

<p>Spaces X for which the annihilator of Sλ(X), the λ-super socle of C(X) (i.e., the set of elements of C(X) that cardinality of their cozerosets are less than λ, where λ is a regular cardinal number such that λ≤|X|) is generated by an idempotent are characterized. This enables us to find a topological property equivalent to essentiality of Sλ(X). It is proved that every prime ideal in C(X) containing Sλ(X) is essential and it is an intersection of free prime ideals. Primeness of Sλ(X) is characterized via a fixed maximal ideal of C(X).</p>

scholarly journals Fuzzy Prime Ideal Theorem in Residuated Lattices

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Pierre Carole Kengne ◽  
Blaise Blériot Koguep ◽  
Celestin Lele
Keyword(s):  
Prime Ideal ◽  
Maximal Ideal ◽  
Residuated Lattice ◽  
Residuated Lattices ◽  
Prime Ideals ◽  
Fuzzy Subset ◽  
Different Types ◽  
Fuzzy Ideal ◽  
Prime Ideal Theorem

This paper mainly focuses on building the fuzzy prime ideal theorem of residuated lattices. Firstly, we introduce the notion of fuzzy ideal generated by a fuzzy subset of a residuated lattice and we give a characterization. Also, we introduce different types of fuzzy prime ideals and establish existing relationships between them. We prove that any fuzzy maximal ideal is a fuzzy prime ideal in residuated lattice. Finally, we give and prove the fuzzy prime ideal theorem in residuated lattice.

scholarly journals NONCOMMUTATIVE HILBERT RINGS

2004 ◽  
Vol 03 (04) ◽  
pp. 437-443 ◽  
Author(s):  
ALGIRDAS KAUCIKAS ◽  
ROBERT WISBAUER
Keyword(s):  
Prime Ideal ◽  
Maximal Ideal ◽  
Commutative Rings ◽  
The Other ◽  
Prime Ideals ◽  
Natural Form ◽  
Noncommutative Rings ◽  
Ideal Formula ◽  
Contraction Property ◽  
Maximal Ideals

Commutative rings in which every prime ideal is the intersection of maximal ideals are called Hilbert (or Jacobson) rings. This notion was extended to noncommutative rings in two different ways by the requirement that prime ideals are the intersection of maximal or of maximal left ideals, respectively. Here we propose to define noncommutative Hilbert rings by the property that strongly prime ideals are the intersection of maximal ideals. Unlike for the other definitions, these rings can be characterized by a contraction property: R is a Hilbert ring if and only if for all n∈ℕ every maximal ideal [Formula: see text] contracts to a maximal ideal of R. This definition is also equivalent to [Formula: see text] being finitely generated as an [Formula: see text]-module, i.e., a liberal extension. This gives a natural form of a noncommutative Hilbert's Nullstellensatz. The class of Hilbert rings is closed under finite polynomial extensions and under integral extensions.

scholarly journals Directed unions of local monoidal transforms and GCD domains

2019 ◽  
Vol 19 (04) ◽  
pp. 2050061
Author(s):  
Lorenzo Guerrieri
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Maximal Ideal ◽  
General Setting ◽  
Regular Local Ring ◽  
Dimension Formula

Let [Formula: see text] be a regular local ring of dimension [Formula: see text]. A local monoidal transform of [Formula: see text] is a ring of the form [Formula: see text], where [Formula: see text] is a regular parameter, [Formula: see text] is a regular prime ideal of [Formula: see text] and [Formula: see text] is a maximal ideal of [Formula: see text] lying over [Formula: see text] In this paper, we study some features of the rings [Formula: see text] obtained as infinite directed union of iterated local monoidal transforms of [Formula: see text]. In order to study when these rings are GCD domains, we also provide results in the more general setting of directed unions of GCD domains.

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