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In this paper, we prove a result similar to results of Itoh [J. Algebra 150 (1992), pp. 101–117] and Hong-Ulrich [J. Lond. Math. Soc. (2) 90 (2014), pp. 861–878], proving that integral closure of an ideal is compatible with specialization by a general element of that ideal for ideals of height at least two in a large class of rings. Moreover, we show integral closure of sufficiently large powers of the ideal is compatible with specialization by a general element of the original ideal. In a polynomial ring over an infinite field, we give a class of squarefree monomial ideals for which the integral closure is compatible with specialization by a general linear form. [ABSTRACT FROM AUTHOR]

Let S be the polynomial ring on the space of non-square generic matrices or the space of odd-sized skew-symmetric matrices, and let I be the determinantal ideal of maximal minors or Pf the ideal of sub-maximal Pfaffians, respectively. Using desingularizations and representation theory of the general linear group we expand upon work of Raicu–Weyman–Witt [Adv. Math. 250 (2014), pp. 596–610] to determine the S-module structures of Ext^j_S(S/I^t, S) and Ext^j_S(S/Pf^t, S), from which we get the degrees of generators of these Ext modules. As a consequence, via graded local duality we answer a question of Wenliang Zhang [J. Pure Appl. Algebra 225 (2021), Paper No. 106789] on the socle degrees of local cohomology modules of the form H^j_\mathfrak {m}(S/I^t). [ABSTRACT FROM AUTHOR]

We resolve an open question posed by Nagpal, Sam and Snowden [Selecta. Math. (N.S.) 22 (2016), pp. 913–937] in 2015 concerning a Gröbner theoretic approach to the noetherianity of the twisted commutative algebra Sym(Sym^2(\mathbf {C}^\infty)). We provide a negative answer to their question by producing an explicit antichain. In doing so, we establish a connection to well-studied posets of graphs under the subgraph and induced subgraph relation. We then analyze this connection to suggest future paths of investigation. [ABSTRACT FROM AUTHOR]

In this paper, we study conditions guaranteeing that a product of ideals defines a Golod ring. We show that for a 3-dimensional regular local ring (or 3-variable polynomial ring) (R, \mathfrak {m}), the ideal I \mathfrak {m} always defines a Golod ring for any proper ideal I \subset R. We also show that non-Golod products of ideals are ubiquitous; more precisely, we prove that for any proper ideal with grade \geqslant 4, there exists an ideal J \subseteq I such that IJ is not Golod. We conclude by showing that if I is any proper ideal in a 3-dimensional regular local ring and \mathfrak {a} \subseteq I a complete intersection, then \mathfrak {a} I is Golod. [ABSTRACT FROM AUTHOR]

The notion of initial ideal for an ideal of a polynomial ring appears in the theory of Gröbner 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 polynomial ring, which is used in the theory of SAGBI (Subalgebra Analogue to Gröbner Bases for Ideals) basis. The initial algebra of a finitely generated subalgebra is not always finitely generated, and no general criterion for finite generation is known. The aim of this paper is to present a new class of finitely generated subalgebras having non-finitely generated initial algebras. The class contains a subalgebra for which the set of initial algebras is uncountable, as well as a subalgebra with finitely many distinct initial algebras. [ABSTRACT FROM AUTHOR]

Let D be a Dedekind domain with finite residue fields. We provide topological insights into certain classes of ideals of Int(D) lying over a given maximal ideal \mathfrak m of D. We completely determine invertible/divisorial ideals in terms of topological properties of subsets of the \mathfrak m-adic completion of D. Moreover, these results are naturally extended to overrings of Int(D). As an application we provide explicit constructions of divisorial ideals of Int(D) which are not finitely generated. [ABSTRACT FROM AUTHOR]

Say that (x, y, z) is a positive primitive integral Pythagorean triple if x, y, z are positive integers without common factors satisfying x^2 + y^2 = z^2. An old theorem of Berggren gives three integral invertible linear transformations whose semi-group actions on (3, 4, 5) and (4, 3, 5) generate all positive primitive Pythagorean triples in a unique manner. We establish an analogue of Berggren's theorem in the context of a one-variable polynomial ring over a field of characteristic \neq 2. As its corollaries, we obtain some structure theorems regarding the orthogonal group with respect to the Pythagorean quadratic form over the polynomial ring. [ABSTRACT FROM AUTHOR]

Let R be a finitely generated \mathbb N-graded algebra domain over a Noetherian ring and let I be a homogeneous ideal of R. Given P\in Ass(R/I) one defines the v-invariant v_P(I) of I at P as the least c\in \mathbb N such that P=I:f for some f\in R_c. A classical result of Brodmann [Proc. Amer. Math. Soc. 74 (1979), pp. 16–18] asserts that Ass(R/I^n) is constant for large n. So it makes sense to consider a prime ideal P\in Ass(R/I^n) for all the large n and investigate how v_P(I^n) depends on n. We prove that v_P(I^n) is eventually a linear function of n. When R is the polynomial ring over a field this statement has been proved independently also by Ficarra and Sgroi in their recent preprint [ Asymptotic behaviour of the \text {v}-number of homogeneous ideals , https://arxiv.org/abs/2306.14243, 2023]. [ABSTRACT FROM AUTHOR]

Let I be an equidimensional ideal of a ring polynomial R over \mathbb {C} and let J be its generic linkage. We prove that there is a uniform bound of the difference between the F-pure thresholds of I_p and J_p via the generalized Frobenius powers of ideals. This provides evidence that the F-pure threshold of an equidimensional ideal I is less than that of its generic linkage. As a corollary we recover a result on log canonical thresholds of generic linkage by Niu. [ABSTRACT FROM AUTHOR]

Let R=k[x,y] be a polynomial ring over a field k of prime characteristic p and let E denote the injective hull of k (which is isomorphic to H^2_{(x,y)}(R)). We prove that E is not an injective object in the category of graded F-modules over R. This answers in the negative a question raised by Lyubeznik-Singh-Walther [J. Eur. Math. Soc. (JEMS) 18 (2016), pp. 2545- 2578]. [ABSTRACT FROM AUTHOR]

We determine the Grothendieck rings of the category of finite-dimensional modules over Queer Lie superalgebras via their rings of characters. In particular, we show that the \mathbb {Q}-span of the ring of characters of the Queer Lie supergroup Q(n) is isomorphic to the ring of Laurent polynomials in x_{1},\ldots,x_{n} for which the evaluation x_{n-1}=-x_{n}=t is independent of t. We thus complete the description of Grothendieck rings for all classical Lie superalgebras. [ABSTRACT FROM AUTHOR]

Poonen proved an analogue for Drinfeld modules of the Mordell-Weil theorem. We shall generalize his results to the case of t-modules which have the height property. Moreover, we also prove a Mordell–Weil theorem for certain t-modules which lack the height property. In particular, let A be the polynomial ring in one variable \theta over the finite field \mathbb {F}_q with the quotient field K. Let (E, \phi) be an iterated extension of the tensor powers of the Carlitz module defined over K. We show that the \mathbb {F}_q[t]-module E(K) is isomorphic to the direct sum of its torsion module, which is finite, with a free \mathbb {F}_q[t]-module of rank \aleph _0. [ABSTRACT FROM AUTHOR]

We present an infinite family of recursive formulas that count binary integer partitions satisfying natural divisibility conditions and show that these counts are interrelated via partial sums. Moreover, we interpret the partitions we study in the language of graded polynomial rings and apply this to the mod 2 Steenrod algebra to compute the free rank of certain homology modules in stable homotopy theory. [ABSTRACT FROM AUTHOR]

In the classical theory, a famous by-product of the continued fraction expansion of quadratic irrational numbers \sqrt {D} is the solution to Pell's equation for D. It is well-known that, once an integer solution to Pell's equation exists, we can use it to generate all other solutions (u_n,v_n)_{n\in \mathbb {Z}}. Our object of interest is the polynomial version of Pell's equation, where the integers are replaced by polynomials with complex coefficients. We then investigate the factors of v_n(t). In particular, we show that over the complex polynomials, there are only finitely many values of n for which v_n(t) has a repeated root. Restricting our analysis to \mathbb {Q}[t], we give an upper bound on the number of "new" factors of v_n(t) of degree at most N. Furthermore, we show that all "new" linear rational factors of v_n(t) can be found when n \leq 3, and all "new" quadratic rational factors when n \leq 6. [ABSTRACT FROM AUTHOR]

Given a field k and a finite group G, the Beckmann–Black problem asks whether every Galois field extension F/k with group G is the specialization at some t_0 \in k of some Galois field extension E/k(T) with group G and E \cap \overline {k} = k. We show that the answer is positive for arbitrary k and G, if one waives the requirement that E/k(T) is normal. In fact, our result holds if \operatorname {Gal}(F/k) is any given subgroup H of G and, in the special case H=G, we provide a similar conclusion even if F/k is not normal. We next derive that, given a division ring H and an automorphism \sigma of H of finite order, all finite groups occur as automorphism groups over the skew field of fractions H(T, \sigma) of the twisted polynomial ring H[T, \sigma ]. [ABSTRACT FROM AUTHOR]

We use the k[V]-module generator of the dual module of the polynomial ring k[V] over its subring of invariants of a finite group to construct modular invariants and show that it behaves better than the transfer homomorphism. [ABSTRACT FROM AUTHOR]