Related papers: Finitely generated powers of prime ideals

A famous result due to I. M. Isaacs states that if a commutative ring $R$ has the property that every prime ideal is principal, then every ideal of $R$ is principal. This motivates ring theorists to study commutative rings for which every…

In this paper, we introduce the notion of the strong Rees property (SRP) for $\mathfrak{m}$-primary ideals of a Noetherian local ring and prove that any power of the maximal ideal $\mathfrak{m}$ has its property if the associated graded…

Let $K$ be a field and let $A$ be a finitely generated prime $K$-algebra. We generalize a result of Smith and Zhang, showing that if $A$ is not PI and does not have a locally nilpotent ideal, then the extended centre of $A$ has…

It is shown that any finitely generated subring of a global field has a universal first-order definition in its fraction field. This covers Koenigsmann's result for the ring of integers and its subsequent extensions to rings of integers in…

It is proved that if $D$ is a $UFD$ and $R$ is a $D$-algebra, such that $U(R)\cap D\neq U(D)$, then $R$ has a maximal subring. In particular, if $R$ is a ring which either contains a unit $x$ which is not algebraic over the prime subring of…

Given a symbolic power of a homogeneous ideal in a polynomial ring, we study the problem of determining which powers of the ideal contain it. For ideals defining 0-dimensional subschemes of projective space, as an immediate corollary of our…

We prove that a commutative parasemifield S is additively idempotent provided that it is finitely generated as a semiring. Consequently, every proper commutative semifield T that is finitely generated as a semiring is either additively…

Let $G$ be a finitely generated abelian group, and let $S = A[x_1, ..., x_n]$ be a $G$-graded polynomial ring over a commutative ring $A$. Let $I_1, ..., I_s$ be $G$-homogeneous ideals in $S$, and let $M$ be a finitely generated $G$-graded…

It is shown that a ring is left semihereditary if and only each homomorphic image of its injective hull as left module is FP-injective. It is also proven that a commutative ring R is reduced and arithmetical if and only if E/U if…

We prove that if $R$ is a commutative Noetherian local pro-$p$domain of characteristic 0 then every finitely generated$R$-standard group is $R$-linear.

In this paper, new and significant advances on the understanding the structure of p.p. rings and their generalizations have been made. Especially among them, it is proved that a commutative ring $R$ is a generalized p.p. ring if and only if…

Let $ A \subset B$ be rings. An ideal $ J \subset B$ is called power stable in $A$ if $ J^n \cap A = (J\cap A)^n$ for all $ n\geq 1$. Further, $J$ is called ultimately power stable in $A$ if $ J^n \cap A = (J\cap A)^n$ for all $n$ large…

The ring of periodic distributions on ${\mathbb{R}}^{\tt d}$ with usual addition and with convolution is considered. Via Fourier series expansions, this ring is isomorphic to the ring ${\mathcal{S}}'({\mathbb{Z}}^{\tt d})$ of all maps…

Classes of algebraic structures that are defined by equational laws are called varieties or equational classes. A variety is finitely generated if it is defined by the laws that hold in some fixed finite algebra. We show that every…

For any prime number p and any positive real number {\alpha}, we construct a finitely generated group {\Gamma} with p-gradient equal to {\alpha}. This construction is used to show that there exist uncountably many pairwise non-commensurable…

The notion of initial ideal for an ideal of a polynomial ring appears in the theory of Gr\"obner basis. Similarly to the initial ideals, we can define the initial algebra for a subalgebra of a polynomial ring, or more generally of a Laurent…

We study conditions on polynomials such that the ideal generated by their orbits under the symmetric group action becomes a monomial ideal or has a monomial radical. If the polynomials are homogeneous, we expect that such an ideal has a…

Let $R$ be a commutative Noetherian ring, $\fa$ an ideal of $R$ and $M$ a finitely generated $R$--module. Let $t$ be a non-negative integer such that $\H^i_\fa(M)$ is $\fa$--cofinite for all $i<t$. It is well--known that…

Let p be a prime and q=p^g. We show that the Grothendieck ring of finitely generated F_{q}[SL(2,F_{q})]-modules is naturally isomorphic to the quotient of the polynomial algebra Z[x] by the ideal generated by f^[g](x)-x, where…

We introduce the theory of monoidal Groebner bases, a concept which generalizes the familiar notion in a polynomial ring and allows for a description of Groebner bases of ideals that are stable under the action of a monoid. The main…