Narrow Results

Constructions are given of Noetherian maximal orders that are finitely presented algebras over a field K, defined by monomial relations. In order to do this, it is shown that the underlying homogeneous information determines the algebraic structure of the algebra. So, it is natural to consider such algebras as semigroup algebras K[S] and to investigate the structure of the monoid S. The relationship between the prime ideals of the algebra and those of the monoid S is one of the main tools. Results analogous to fundamental facts known for the prime spectrum of algebras graded by a finite group are obtained. This is then applied to characterize a large class of prime Noetherian maximal orders that satisfy a polynomial identity, based on a special class of submonoids of polycyclic-by-finite groups. The main results are illustrated with new constructions of concrete classes of finitely presented algebras of this type.

We obtain the explicit generators of the annihilator ideal for the tensor product of any finite dimensional simple module over quantum group , by using the weight property of ideals in when q is not a root of unity. As an application, we give a presentation of quantum generalized Schur algebra.

This paper is concerned with the study of the dimension theory of tensor products of algebras over a field k. We answer an open problem set in [6] and compute dim(A ⊗_k B) when A is a k-algebra arising from a specific pullback construction involving AF-domains and B is an arbitrary k-algebra. On the other hand, we deal with the question (Q) set in [5] and show, in particular, that such a pullback A is in fact a generalized AF-domain.

The purpose of this paper is to explore new aspects of the prime ideal structure of tensor products of algebras over a field k. We prove that given a k-algebra A and a normal field extension K of k (in the sense of Galois theory), then for any prime ideals P₁ and P₂ of K ⊗_k A lying over a fixed prime ideal p of A, there exists a k-automorphism σ of K such that (σ ⊗_kid_A)(P₁) = P₂. As an Application, we establish a result related to the dimension theory of tensor products stating that, for two arbitrary k-algebras A and B, the minimal prime ideals of p ⊗_k B + Aσ_k q have the same height, for any prime ideals p and q of A and B, respectively.

A (commutative integral) domain R is said to be valuative if, for each nonzero element u in the quotient field of R, at least one of R ⊆ R[u] and R ⊆ R[u^-1] has no proper intermediate rings. Such domains are closely related to valuation domains. If R is a valuative domain, then R has at most three maximal ideals, and at most two if R is not integrally closed. Also, if R is valuative, the set of nonmaximal prime ideals of R is linearly ordered, at most one maximal ideal of R does not contain each nonmaximal prime of R, and R_P is a valuation domain for each prime P except for at most one maximal ideal. Any integrally closed valuative domain is a Bézout domain. Valuation domains are characterized as the quasilocal integrally closed valuative domains. Each one-dimensional Prüfer domain with at most three maximal ideals is valuative.

In this paper we study partial actions of abelian groups on a ring R having an enveloping action. Among other results we prove that if α is such a partial action and (T, β) its globalization, then there exists a one-to-one correspondence between closed (respectively, R-disjoint prime) ideals of R ⋆_α G and closed (respectively, T-disjoint prime) ideals of T ⋆_β G. We also prove that there exists a one-to-one correspondence between closed (respectively, R-disjoint prime) ideals of R ⋆_α G and closed (respectively, -disjoint prime) ideals of , where is the left α-quotient ring of R. Finally, we use these results to study strongly prime ideals and nonsingular prime ideals of R ⋆_α G.

Let R ⊂ S be an extension of integral domains. We say that (R, S) is a well-centered pair of rings, if each intermediate ring T between R and S is well-centered on R, in the sense that each principal ideal of T is generated by an element of R. The aim of this paper is to study well-centered pairs of rings. We investigate the structure of the intermediate rings T between R and S that are well-centered on R. We establish the relationship between well-centered pairs and normal pairs. We present some examples and counterexamples illustrating our theory and showing the limits of our results.

Let $R\subset S$ be a (unital) extension of (commutative) rings. We say that $R$ is a maximal non-quasi-local (respectively, non-PVD) subring of $S$ if $R$ is not quasi-local (respectively, PVD) and each subring of $S$ properly containing $R$ is quasi-local (respectively, PVD). The aim of this paper is to study this kind of ring extensions and to investigate the structure of the intermediate rings between $R$ and $S$.

Let $R$ be a commutative ring with $1\ne 0$, and let $I$ be a proper ideal of $R$. Recall that $I$ is an $n$-absorbing ideal if whenever $x_1\cdots x_{n+1}\in I$ for $x_1,\dots ,x_{n+1}\in R$, then there are $n$ of the $x_i$’s whose product is in $I$. We define $I$ to be a semi-$n$-absorbing ideal if $x^{n+1}\in I$ for $x\in R$ implies $x^n\in I$. More generally, for positive integers $m$ and $n$, we define $I$ to be an $(m,n)$-closed ideal if $x^m\in I$ for $x\in R$ implies $x^n\in I$. A number of examples and results on $(m,n)$-closed ideals are discussed in this paper.

In this paper, prime as well as primitive Kumjian–Pask algebras ${\mathrm{\text{KP}}}_R(\mathrm{\Lambda })$ of a row-finite $k$-graph $\mathrm{\Lambda }$ over a unital commutative ring $R$ are completely characterized in graph-theoretic and algebraic terms. By applying quotient $k$-graphs, these results describe prime and primitive graded basic ideals of Kumjian–Pask algebras. In particular, when $\mathrm{\Lambda }$ is strongly aperiodic and $R$ is a field, all prime and primitive ideals of a Kumjian–Pask algebra ${\mathrm{\text{KP}}}_R(\mathrm{\Lambda })$ are determined.

A prime ideal $𝔭$ is said to be strongly prime if whenever $𝔭$ contains an intersection of ideals, $𝔭$ contains one of the ideals in the intersection. A commutative ring with this property for every prime ideal is called strongly zero-dimensional. Some equivalent conditions are given and it is proved that a zero-dimensional ring is strongly zero-dimensional if and only if the ring is quasi-semi-local. A ring is called strongly $n$-regular if in each ideal $𝔞$, there is an element $a$ such that $x=ax$ for all $x\in {𝔞}^n$. Connections between the concepts strongly zero-dimensional and strongly $n$-regular are considered.

Let $R$ be a commutative ring with $1\ne 0$. Recall that a proper ideal $I$ of $R$ is called a 2-absorbing ideal of $R$ if $a,b,c\in R$ and $abc\in I$, then $ab\in I$ or $ac\in I$ or $bc\in I$. A more general concept than 2-absorbing ideals is the concept of $n$-absorbing ideals. Let $n\ge 1$ be a positive integer. A proper ideal $I$ of $R$ is called an n-absorbing ideal of $R$ if $a_1,a_2,\dots ,a_{n+1}\in R$ and $a_1,a_2\cdots a_{n+1}\in I$, then there are $n$ of the $a_i$’s whose product is in $I$. The concept of $n$-absorbing ideals is a generalization of the concept of prime ideals (note that a prime ideal of $R$ is a 1-absorbing ideal of $R$). Let $m$ and $n$ be integers with $1\le n<m$. A proper ideal $I$ of $R$ is called an $(m,n)$-closed ideal of $R$ if whenever $a^m\in I$ for some $a\in R$ implies $a^n\in I$. Let $A$ be a commutative ring with $1\ne 0$ and $M$ be an $A$-module. In this paper, we study $n$-absorbing ideals and $(m,n)$-closed ideals in the trivial ring extension of $A$ by $M$ (or idealization of $M$ over $A$) that is denoted by $A(+)M$.

The aim of the paper is to prove two conjectures from the paper [V. V. Bavula, The algebra of integro-differential operators on a polynomial algebra, J. London Math. Soc. (2) 83 (2011) 517–543, arXiv:math.RA/0912.0723] that the (left and right) global dimension of the algebra ${𝕀}_n:=K〈x_1,\dots ,x_n,\frac{\partial }{\partial x_1},\dots ,\frac{\partial }{\partial x_n},{\int }_1,\dots ,{\int }_n〉$ of polynomial integro-differential operators and the Jacobian algebra ${𝔸}_n$ is equal to $n$ (over a field of characteristic zero). The algebras ${𝕀}_n$ and ${𝔸}_n$ are neither left nor right Noetherian and ${𝕀}_n\subset {𝔸}_n$. Furthermore, they contain infinite direct sums of nonzero left/right ideals and are not domains. An analogue of Hilbert’s Syzygy Theorem is proven for the algebras ${𝕀}_n$, ${𝔸}_n$ and their factor algebras. It is proven that the global dimension of all prime factor algebras of the algebras ${𝕀}_n$ and ${𝔸}_n$ is $n$ and the weak global dimension of all the factor algebras of ${𝕀}_n$ and ${𝔸}_n$ is $n$.

Let $R$ be a commutative ring with nonzero identity. In this paper, we introduce the concept of 1-absorbing primary ideals in commutative rings. A proper ideal $I$ of $R$ is called a $1$-absorbing primary ideal of $R$ if whenever nonunit elements $a,b,c\in R$ and $abc\in I$, then $ab\in I$ or $c\in \sqrt{I}.$ Some properties of 1-absorbing primary ideals are investigated. For example, we show that if $R$ admits a 1-absorbing primary ideal that is not a primary ideal, then $R$ is a quasilocal ring. We give an example of a 1-absorbing primary ideal of $R$ that is not a primary ideal of $R$. We show that if $R$ is a Noetherian domain, then $R$ is a Dedekind domain if and only if every nonzero proper 1-absorbing primary ideal of $R$ is of the form $P^n$ for some nonzero prime ideal $P$ of $R$ and a positive integer $n\ge 1$. We show that a proper ideal $I$ of $R$ is a 1-absorbing primary ideal of $R$ if and only if whenever $I_1{I}_2{I}_3\subseteq I$ for some proper ideals $I_1,I_2,I_3$ of $R$, then $I_1{I}_2\subseteq I$ or $I_3\subseteq \sqrt{I}.$

Let $R$ be a commutative ring with identity. In this paper, we introduce the concept of $1$-absorbing prime ideals which is a generalization of prime ideals. A proper ideal $I$ of $R$ is called $1$-absorbing prime if for all nonunit elements $a,b,c\in R$ such that $abc\in I$, then either $ab\in I$ or $c\in I$. Some properties of $1$-absorbing prime are studied. For instance, it is shown that if $R$ admits a $1$-absorbing prime ideal that is not a prime ideal, then $R$ is a quasi–local ring. Among other things, it is proved that a proper ideal $I$ of $R$ is $1$-absorbing prime if and only if the inclusion $I_1{I}_2{I}_3\subseteq I$ for some proper ideals $I_1,I_2,I_3$ of $R$ implies that $I_1{I}_2\subseteq I$ or $I_3\subseteq I$. Also, $1$-absorbing prime ideals of PIDs, valuation domains, Prufer domains and idealization of a modules are characterized. Finally, an analogous to the Prime Avoidance Theorem and some applications of this theorem are given.

If $A\subseteq B$ are (commutative) rings, $[A,B]$ denotes the set of intermediate rings and $(A,B)$ is called an almost valuation (AV)-ring pair if each element of $[A,B]$ is an AV-ring. Many results on AV-domains and their pairs are generalized to the ring-theoretic setting. Let $R\subseteq S$ be rings, with ${\overline{R}}_S$ denoting the integral closure of $R$ in $S$. Then $(R,S)$ is an AV-ring pair if and only if both $(R,{\overline{R}}_S)$ and $({\overline{R}}_S,S)$ are AV-ring pairs. Characterizations are given for the AV-ring pairs arising from integrally closed (respectively, integral; respectively, minimal) ring extensions $R\subseteq S$. If $(R,S)$ is an AV-ring pair, then $R\subseteq S$ is a P-extension. The AV-ring pairs $(R,S)$ arising from root extensions are studied extensively. Transfer results for the “AV-ring” property are obtained for pullbacks of $(B,I,D)$ type, with applications to pseudo-valuation domains, integral minimal ring extensions, and integrally closed maximal non-AV subrings. Several sufficient conditions are given for $(R,S)$ being an AV-ring pair to entail that $S$ is an overring of $R$, but there exist domain-theoretic counter-examples to such a conclusion in general. If $(R,S)$ is an AV-ring pair and $R\subseteq S$ satisfies FCP, then each intermediate ring either contains or is contained in ${\overline{R}}_S$. While all AV-rings are quasi-local going-down rings, examples in positive characteristic show that an AV-domain need not be a divided domain or a universally going-down domain.

Let $R$ be a commutative ring, we say that ${𝒜}\subseteq \mathrm{\text{Spec}}(R)$ has prime avoidance property, if $I\subseteq {\bigcup }_{P\in {𝒜}}P$ for an ideal $I$ of $R$, then there exists $P\in {𝒜}$ such that $I\subseteq P$. We exactly determine when ${𝒜}\subseteq \mathrm{\text{Spec}}(R)$ has prime avoidance property. In particular, if ${𝒜}$ has prime avoidance property, then ${𝒜}$ is compact. For certain classical rings we show the converse holds (such as Bezout rings, $QR$-domains, zero-dimensional rings and $C(X)$). We give an example of a compact set ${𝒜}\subseteq \mathrm{\text{Spec}}(R)$, where $R$ is a Prufer domain, which has not prime avoidance property. Finally, we show that if $V,V_1,\dots ,V_n$ are valuation domains for a field $K$ and $V[x]⊈{\bigcup }_{i=1}^n{V}_i$ for some $x\in K$, then there exists $v\in V$ such that $v+x\notin {\bigcup }_{i=1}^n{V}_i$.

The main purpose of this paper is to answer a question which was left open in N. Jarboui and S. Aljubran [Maximal non-integrally closed subrings of an integral domain, Ric. Mat. (2020), doi.org/10.1007/s11587-020-00500-0] asking for a characterization of maximal non-integrally closed subrings of arbitrary rings.

We continue a very fruitful line of inquiry into the multiplicative ideal theory of an arbitrary Leavitt path algebra $L$. Specifically, we show that factorizations of an ideal in $L$ into irredundant products or intersections of finitely many prime-power ideals are unique, provided that the ideals involved are powers of distinct prime ideals. We also characterize the completely irreducible ideals in $L$, which turn out to be prime-power ideals of a special type, as well as ideals that can be factored into products or intersections of finitely many completely irreducible ideals.

The work of Helmer [Divisibility properties of integral functions, Duke Math. J. 6(2) (1940) 345–356] applied algebraic methods to the field of complex analysis when he proved the ring of entire functions on the complex plane is a Bezout domain (i.e. all finitely generated ideals are principal). This inspired the work of Henriksen [On the ideal structure of the ring of entire functions, Pacific J. Math. 2(2) (1952) 179–184. On the prime ideals of the ring of entire functions, Pacific J. Math. 3(4) (1953) 711–720] who proved a correspondence between the maximal ideals within the ring of entire functions and ultrafilters on sets of zeroes as well as a correspondence between the prime ideals and growth rates on the multiplicities of zeroes. We prove analogous results on rings of analytic functions in the non-Archimedean context: all finitely generated ideals in the ring of analytic functions on an annulus of a characteristic zero non-Archimedean field are two-generated but not guaranteed to be principal. We also prove the maximal and prime ideal structure in the non-Archimedean context is similar to that of the ordinary complex numbers; however, the methodology has to be significantly altered to account for the failure of Weierstrass factorization on balls of finite radius in fields which are not spherically complete, which was proven by Lazard [Les zeros d’une function analytique d’une variable sur un corps value complet, Publ. Math. l’IHES 14(1) (1942) 47–75].