On regular rings and self-injective rings

Regular, right self-injective rings are considered. We settle the question of when such a ring is a right V-ring, i.e., when each simple right module over the ring is injective. It is proved that a regular, right self-injective V-ring of the power of the continuum has a bounded nilpotency index.

According to [Comm. Algebra 46 (2018), no. 2, 870–886], a ring R R is called right Utumi-ring ( U U -ring) if, whenever A A and B B are right ideals of R R with A ≅ B A\cong B and A ∩ B = 0 , A\cap B=0, there exist idempotents e e , f f and g g of R R such that A ⊆ e s s e R A\subseteq ^{ess}eR , B ⊆ e s s f R B\subseteq ^{ess}fR and e R ⊕ f R = g R eR\oplus fR=gR . The class of U U -rings is a strict and simultaneous generalization of quasi-continuous, square-free and automorphism-invariant rings. In this paper, we prove that every right U U -ring that is totally non-abelian (i.e. every non-zero right ideal of R R contains a non-central idempotent) is generated as a ring by its idempotents. Moreover, under these assumptions R R turns out to be regular and right self-injective. This result extends the work of Utumi who proved that every totally non-abelian regular right self-injective ring is generated as a ring by its idempotents.

One of the main results of this paper is the characterization of left FP-injective rings as rings over which every finitely presented right module is torsionless. The necessity of the torsionless condition (in fact reflexivity) previously has been observed for special classes of FP-injective rings (e.g., self-injective rings (Kato and McRae), right and left coherent and self FP-injective rings (Stenström)). Furthermore, if the ring is left coherent, it is equivalent to say that every finitely presented right module is reflexive. We investigate the condition (called IF) that every injective right module is flat and characterize this via FP-injectivity. IF-rings generalize regular rings and QF-rings, and under certain chain or homological conditions coincide with regular or QF-rings (e.g., w g l dim ⁡ R ≦ 1 ⇒ regular wgl\dim R \leqq 1 \Rightarrow {\text {regular}} , noetherian, or right perfect ⇒ QF) {\text {right perfect}} \Rightarrow {\text {QF)}} .

This note states necessary and sufficient conditions for a ring to be a right order in certain self-injective rings. A ring R is said to have the dense extension property if each R-homomorphism from a right ideal of R into R can be lifted to some dense right ideal of R. A right ideal K is rationally closed if for each $x \in R - K$ the set ${x^{ - 1}}K = \{ y \in R:xy \in K\}$ is not a dense right ideal of R. We state a major result. Let $\dim R$ denote the Goldie dimension of a ring R and $Z(R)$ the right singular ideal of R. Then R is a right order in a self-injective cogenerator ring if and only if R has the dense extension property, $Z(R)$ is rationally closed and the factor ring $R/Z(R)$ is semiprime with $\dim R/Z(R) = \dim R < \infty$.

On the structure of selfinjective regular rings

This note is a natural sequel to [8] and [9]. Further characteristic properties of arbitrary von Neumann regular rings and strongly regular rings are given in terms of annihilators and simple modules. A prime ring with certain annihilator conditions is shown to be primitive (this is related to the following problem ofKaplansky: Are prime regular rings primitive?). Necessary and sufficient conditions for leftq-rings to be regular are also considered: For example, a leftq-ring is regular iff every simple rightA-module is flat. A sufficient condition is given for a leftqc-ring to be a uniserial, strongly left and strongly rightqc, left and rightq-ring. One of the main results ofJain, Mohamed andSingh onq-rings [5, Theorem 2.13] is generalised. Finally, it is shown that a prime left continuous ring either has zero socle or is primitive, left self-injective regular.

The local factorization theorem of Zariski and Abhyankar characterizes all $2$-dimensional regular local rings which lie between a given $2$-dimensional regular local ring $R$ and its quotient field as finite quadratic transforms of $R$. This paper shows that every regular local ring $R$ of dimension $n > 2$ has infinitely many minimal regular local overrings which cannot be obtained by a monoidal transform of $R$. These overrings are localizations of rings generated over $R$ by certain quotients of elements of an $R$-sequence. Necessary and sufficient conditions are given for this type of extension of $R$ to be regular.

An element x of a semigroup [ring] A is said to regular if there is an element y of A such that x = xyx, and A is called a regular semigroup [(Von Neumann) regular ring] if every element of A is regular. A quasi-ideal of a semigroup [ring] A is a subsemigroup [subring] Q of A such that AQ intersection QA [is a subset of] Q,and a bi-ideal of A is a subsemingroup [subring] B of A such that BAB [is a subset of] B. Recall that for nonempty subsets X and Y of a ring A, XY denotes the set of all finite sums of the form sigma x[subscript i] y[subscript i] where x[subscript i] [is an element of set] X and Y[subscript i] [is an element of set] Y. We say that a semigroup or a ring has the BQ-property if its quasi-ideals and bi-ideals coincide. It is know that every regular semigroup and every regular ring has the BQ-property. For a nonempty set X, let T(X) denote the full transformation semigroup on X, and for [is an empty set] is not equal to Y[is a subset of] X, let T (X,Y) and T [bar] (X, Y) be the subsemigroups of T(X) defined by T(X, Y)={ alpha is an element of a set T(X) l ran alpha is a subset of Y} and T[bar] (X, Y)={ alpha is an elemaent of a set T(X) l Yalpha is a subset of Y}. Symons and Magill introduced and studied T(X, Y) and T[bar] (X, Y) in 1975 and 1966, respectively. If V is a vector space over a field F, let L[subscript F](V) be the set of all linear transformations alpha : V vector V. For a subspace W of V, define L[subscript F] (V, W) and L[bar][subscript F] (V, W) analogously as follows : L[subscript F] (V, W) = {alpha is an element of a set L[subscriptF(V) l ran alpha is a subset of W} and L[bar][subscript F](V, W) = {alpha is an element of a set L[subscritp F] (V) l { Walpha is a subset of W},and we also consider K[subscript F](V, W) = {alpha L[subscriptF] (V) l W is a subset of ker alpha}. Then L[subscriptF](V, W), L[bar] [subscript F](V, W) and K[subscript F] (V, W) are subsemigroups of (L[subscript F] (V), omicron) and subrings of (L[subscript F] (V),+,omicron) where omicron and+are the composition and usual addition of linear transformations, respectively. This research consists of two major parts. In the first part, we give necessary and sufficient conditions for the elements of these semigroups to be regular. As a consequence, characterizations determinging when these semigroups are regular are given. In the second part, we provide necessary and sufficient conditions for these semigroups and rings to have the BQ-property.

On regular self-injective rings

Let V be a complete regular local (Noetherian) ring and let f ∈ V be a nonunit. We find necessary and sufficient conditions for to be the completion with respect to the maximal ideal of an integral domain of the form where S is a regular local ring whose completion with respect to its maximal ideal is V. In addition, if contains the rationals, we give necessary and sufficient conditions for to be the completion of an excellent local domain of the form , where f ∈ S, and S is a regular local ring whose completion is V.

In this paper, it is shown that the value semigroup of a prime divisor of the second kind on a 2 2 -dimensional regular local ring is symmetric. Further, a necessary and sufficient condition for two prime divisors of the second kind on a 2 2 -dimensional regular local ring to have the same value semigroup is obtained.

Contractedness of 𝔪-primary integrally closed ideals played a central role in the development of Zariski's theory of integrally closed ideals in two-dimensional regular local rings (R, 𝔪). In such rings, the contracted 𝔪-primary ideals are known to be characterized by the property that I: 𝔪 = I: x for some x ∈ 𝔪 ∖𝔪2. We call the ideals with this property full ideals and compare this class of ideals with the classes of 𝔪-full ideals, basically full ideals, and contracted ideals in higher dimensional regular local rings. The 𝔪-full ideals are easily seen to be full. In this article, we find a sufficient condition for a full ideal to be 𝔪-full. We also show the equivalence of the properties full, 𝔪-full, contracted, integrally closed, and normal, for the class of parameter ideals. We then find a sufficient condition for a basically full parameter ideal to be full.

Valuations with preassigned proximity relations

Let be a ring, and the left and right annihilators of the element , the two-sided ideal in called the additive controller, and let be an -isomorphism (i.e., multiplicative isomorphism) and its defect. An ideal in the ring is called an -ideal if for all -isomorphisms , is an ideal in and if and only if . It is shown that Very general sufficient conditions are given that a multiplicative isomorphism of subsemigroups of multiplicative semigroups of rings be extendible to the isomorphism of the subrings generated by them. Minimal prime ideals and the prime radical of a ring are -ideals. The strongly regular and regular rings that have unique addition are characterized.Bibliography: 29 titles.

Three notes on the order of ideals defining hypersurfaces