Pseudo-valuation Ring Research Articles

Maximal non-ϕ-pseudo-valuation rings

The notion of maximal non-[Formula: see text]-pseudo-valuation ring is introduced which generalizes the concept of maximal non-pseudo-valuation domain. The equivalence of maximal non-[Formula: see text]-PVR and maximal non-local ring is established under some conditions. The concept of maximal non-PVR is also introduced and it is shown that a maximal non-PVR is either a [Formula: see text]-PVR or a maximal non-[Formula: see text]-PVR. Examples are also given to show that all the introduced concepts are independent of each other, in general.

Amalgamated Algebras Issued from $$\phi $$-Chained Rings and $$\phi $$-Pseudo-Valuation Rings

Let $$f : A \longrightarrow B$$ be a ring homomorphism and let J be an ideal of B. In this article, we study the possible transfer of the properties of being a $$\phi $$ -ring, a $$\phi $$ -chained ring, and a $$\phi $$ -pseudo-valuation ring to the amalgamated algebra $$A\bowtie ^{f}J$$ . Our aim is to provide examples of new classes of commutative rings satisfying the aforementioned properties.

Graded pseudo-valuation domains

We introduce a graded version of pseudo-valuation rings for integral domains graded by a torsionless monoid, and we extend several results to the graded situation.

Trivial extensions satisfying certain valuation-like properties

In this article, we study the possible transfer of the properties of being an almost valuation ring, a pseudo-valuation ring and a pseudo-almost valuation ring between a commutative ring A and (the trivial ring extension of A arising from an A-module E). Also, we extend the notion of an almost pseudo-valuation domain to the context of arbitrary commutative rings (possibly with nontrivial zero-divisors), and we also study the possible transfer of this notion between A and . Our aim is to provide examples of new classes of commutative rings satisfying the above-mentioned properties.

ON ϕ-PSEUDO ALMOST VALUATION RINGS

The purpose of this paper is to introduce a new class of rings that is closely related to the classes of pseudo valuation rings (PVRs) and pseudo-almost valuation domains (PAVDs). A commutative ring R is said to be <TEX>${\phi}$</TEX>-ring if its nilradical Nil(R) is both prime and comparable with each principal ideal. The name is derived from the natural map <TEX>${\phi}$</TEX> from the total quotient ring T(R) to R localized at Nil(R). A prime ideal P of a <TEX>${\phi}$</TEX>-ring R is said to be a <TEX>${\phi}$</TEX>-pseudo-strongly prime ideal if, whenever <TEX>$x,y{\in}R_{Nil(R)}$</TEX> and <TEX>$(xy){\phi}(P){\subseteq}{\phi}(P)$</TEX>, then there exists an integer <TEX>$m{\geqslant}1$</TEX> such that either <TEX>$x^m{\in}{\phi}(R)$</TEX> or <TEX>$y^m{\phi}(P){\subseteq}{\phi}(P)$</TEX>. If each prime ideal of R is a <TEX>${\phi}$</TEX>-pseudo strongly prime ideal, then we say that R is a <TEX>${\phi}$</TEX>-pseudo-almost valuation ring (<TEX>${\phi}$</TEX>-PAVR). Among the properties of <TEX>${\phi}$</TEX>-PAVRs, we show that a quasilocal <TEX>${\phi}$</TEX>-ring R with regular maximal ideal M is a <TEX>${\phi}$</TEX>-PAVR if and only if V = (M : M) is a <TEX>${\phi}$</TEX>-almost chained ring with maximal ideal <TEX>$\sqrt{MV}$</TEX>. We also investigate the overrings of a <TEX>${\phi}$</TEX>-PAVR.

Completely pseudo-valuation rings and their extensions

Recall that a commutative ring R is said to be a pseudo-valuation ring if every prime ideal of R is strongly prime. We define a completely pseudovaluation ring. Let R be a ring (not necessarily commutative). We say that R is a completely pseudo-valuation ring if every prime ideal of R is completely prime. With this we prove that if R is a commutative Noetherian ring, which is also an algebra over Q (the field of rational numbers) and ? a derivation of R, then R is a completely pseudo-valuation ring implies that R[x, ?] is a completely pseudo-valuation ring. We prove a similar result when prime is replaced by minimal prime.

Minimal prime ideals of skew polynomial rings and near pseudo-valuation rings

Let R be a ring. We recall that R is called a near pseudo-valuation ring if every minimal prime ideal of R is strongly prime.

On semi-pseudo-valuation rings and their extensions

Recall that a commutative ring R is said to be a pseudo-valuation ring (PVR) if every prime ideal of R is strongly prime. We say that a commutative Noetherian ring R is semi-pseudo-valuation ring if assassinator of every right ideal (which is uniform as a right R-module) is a strongly prime ideal. We also recall that a prime ideal P of a ring R is said to be divided if it is comparable (under inclusion) to every ideal of R. A ring R is called a divided ring if every prime ideal of R is divided. Let R be a commutative ring, σ an automorphism of R and δ a σ-derivation of R. We say that a prime ideal P of R is δ-divided if it is comparable (under inclusion) to every σ-stable and δ-invariant ideal I of R. A ring R is called a δ-divided ring if every prime ideal of R is δ-divided. We say that a Noetherian ring R is semi-δ-divided ring if assassinator of every right ideal (which is uniform as a right R-module) is δ-divided. Let R be a ring and σ an endomorphism of R. Recall that R is said to be a σ(*)-ring if aσ(a) ∈ P(R) implies that a ∈ P(R), where P(R) is the prime radical of R. With this we prove the following

Ore extensions over pseudo-valuation rings

Let R be a commutative Noetherian Q-algebra (Q is the field of rational numbers). Let δ be a derivation of R and σ be an automorphism of R. Then we prove the following: 1. If R is a Pseudo-valuation ring, then R[x, δ] is also a Pseudo-valuation ring. 2. If R is a divided ring, then R[x, δ] is also a divided ring. 3. If R is a Pseudo-valuation ring, thenR[x, x−1, σ] is also a Pseudo-valuation ring. 4. If R is a divided ring, then R[x, x−1, σ] is also a divided ring. Mathemmatics Subject Classification: Primary 16-XX; Secondary 16P40, 16P50, 16U20

Polynomial Rings over Pseudovaluation Rings

LetRbe a ring. Letσbe an automorphism ofR. We define aσ-divided ring and prove the following. (1) LetRbe a commutative pseudovaluation ring such thatx∉Pfor anyP∈Spec(R[x,σ]). ThenR[x,σ]is also a pseudovaluation ring. (2) LetRbe aσ-divided ring such thatx∉Pfor anyP∈Spec(R[x,σ]). ThenR[x,σ]is also aσ-divided ring. Let nowRbe a commutative NoetherianQ-algebra (Qis the field of rational numbers). Letδbe a derivation ofR. Then we prove the following. (1) LetRbe a commutative pseudovaluation ring. ThenR[x,δ]is also a pseudovaluation ring. (2) LetRbe a divided ring. ThenR[x,δ]is also a divided ring.

Pseudo-Chain Rings and Pseudo-Uniserial Modules

ABSTRACT A right ideal I of a ring R is said to be a comparizer right ideal of R if for any right ideals A, B of R, either A ⊆ B or BI ⊆ A. A ring R (with 1) is called a right pseudo-chain ring if for any nonunit a ∈ R, aR is a comparizer right ideal of R. These rings are generalizations of right chain rings and commutative pseudo-valuation rings. In this paper we study the structure of right pseudo-chain rings. We prove that a ring R containing a non-zero-divisor in the Jacobson radical J(R) is a pseudo-chain ring if and only if R is a local subring of a (unique up to isomorphism) chain ring T with J(T) = J(R). We also introduce the concept of a pseudo-uniserial module. #Communicated by E. Puczylowski.

Remarks on pseudo-valuation rings

A prime ideal P of a ring A is said to be a strongly prime ideal if aP and bA are comparable for all a,b ϵ A. We shall say that a ring A is a pseudo-valuation ring (PVR) if each prime ideal of A is a strongly prime ideal. We show that if A is a PVR with maximal ideal M, then every overring of A is a PVR if and only if M is a maximal ideal of every overring of M that does not contain the reciprocal’of any element of M.We show that if R is an atomic domain and a PVD, then dim(R) ≤ 1. We show that if R is a PVD and a prime ideal of R is finitely generated, then every overring of R is a PVD. We give a characterization of an atomic PVD in terms of the concept of half-factorial domain.

On chained overrings of pseudo-valuation rings

A prime ideal P of a commutative ring R with identity is called strongly prime if aP and bR are comparable for every a, b in R. If every prime ideal of R is strongly prime, then R is called a pseudo-valuation ring. It is well-known that a (valuation) chained overring of a Prufer domain R is of the form RP for some prime ideal P of R.In this paper, we show that this statement is valid for a certain class of chained overrings of a pseudo-valuation ring.