Related papers: Unimodular rows over monoid rings

A rank-$r$ integer matrix $A$ is $\Delta$-modular if the determinant of each $r \times r$ submatrix has absolute value at most $\Delta$. The class of $1$-modular, or unimodular, matrices is of fundamental significance in both integer…

This is the first of a series of papers in which we initiate and develop the theory of reflection monoids, motivated by the theory of reflection groups. The main results identify a number of important inverse semigroups as reflection…

In this paper we study right $S$-Noetherian rings and modules, extending of notions introduced by Anderson and Dumitrescu in commutative algebra to noncommutative rings. Two characterizations of right $S$-Noetherian rings are given in terms…

We develop a new technique for studying monomial ideals in the standard polynomial rings $A[X_1,\ldots,X_d]$ where $A$ is a commutative ring with identity. The main idea is to consider induced ideals in the semigroup ring…

We classify all tilting classes over an arbitrary commutative ring via certain sequences of Thomason subsets of the spectrum, generalizing the classification for noetherian commutative rings by…

In this article, we prove that if $R$ is an affine algebra of dimension $d\geq 4$ over $\overline{\mathbb{F}}_{p}$ and $1/(d-1)! \in R,$ then any unimodular row over $R$ of length $d$ can be mapped to a factorial row by elementary…

Let $R$ be a commutative ring with $1\in R$ and $R^{\ast}$ be the multiplicative group of its units. In 1969, Nagell introduced the exceptional unit $u$ if both $u$ and $1-u$ belong to $R^{\ast}$. Let $\mathbb{Z}_n$ be the ring of residue…

A differential module is a module equipped with a square-zero endomorphism. This structure underpins complexes of modules over rings, as well as differential graded modules over graded rings. We establish lower bounds on the class--a…

This is an overview of the idea of a crossed module. For a group, the triple that consists of the group, its group of automorphisms, and the canonical homomorphism from the group to its group of automorphisms constitutes a crossed module.…

We establish an Excision type theorem for niceness of group structure on the orbit space of unimodular rows of length $n$ modulo elementary action. This permits us to establish niceness for relative versions of results for the cases when $n…

Motivated by the Poisson Dixmier-Moeglin equivalence problem, a systematic study of commutative unitary rings equipped with a {\em biderivation}, namely a binary operation that is a derivation in each argument, is here begun, with an eye…

We study the structure of the commutative multiplicative monoid $\mathbb N_0[x]^*$ of all the non-zero polynomials in $\mathbb Z[x]$ with non-negative coefficients. We show that $\mathbb N_0[x]^*$ is not a half-factorial monoid and is not a…

We introduce the notion of independent sequences with respect to a monomial order by using the least terms of polynomials vanishing at the sequence. Our main result shows that the Krull dimension of a Noetherian ring is equal to the…

A module over a ring $R$ is pure projective provided it is isomorphic to a direct summand of a direct sum of finitely presented modules. We develop tools for the classification of pure projective modules over commutative noetherian rings.…

Let R be a commutative ring with identity. The concept of second submodule of an R-module (as a dual notion of prime submodules) was introduced and studied by S.Yassemi in 2001. This notion has obtained a great attention by many authors and…

The first part of the paper will describe a recent result of K. Retert in (\cite{Ret}) for $k[x_1,\ldots,x_n]$ and $k[[x_1,\ldots,x_n]]$. This result states that if $\mathfrak{D}$ is a set of commute $k$-derivations of $k[x,y]$ such that…

The purpose of this note is to introduce a multiplication on the set of homogeneous polynomials of fixed degree d, in a way to provide a duality theory between monomial ideals of K[x_1,\ldots,x_d] generated in degrees \leq n and block…

Let $R$ be a commutative ring with identity and $S$ a multiplicative subset of $R$. In this paper, we introduce and study the notions of $S$-pure $S$-exact sequences and $S$-absolutely pure modules which extend the classical notions of pure…

When does a Noetherian commutative ring $R$ have uniform symbolic topologies on primes--read, when does there exist an integer $D>0$ such that the symbolic power $P^{(Dr)} \subseteq P^r$ for all prime ideals $P \subseteq R$ and all $r >0$?…

The main goal of the paper is the discussion of a deeper interaction between matrix theory over polynomial rings over a field and typical methods of commutative algebra and related algebraic geometry. This is intended in the sense of…