Let $K$ be a field, $A=K[X_1,\cdots ,X_n]$ and $𝕄$ the set of monomials of $A$. It is well known that the set of monomial ideals of $A$ is in a bijective correspondence with the set of all subsemiflows of the $𝕄$-semiflow $𝕄$. We generalize this to the case of term ideals of $A=R[X_1,\cdots ,X_n]$, where $R$ is a commutative Noetherian ring. A term ideal of $A$ is an ideal of $A$ generated by a family of terms $c{X}_1^{{\mu }_1}\cdots X_n^{{\mu }_n}$, where $c\in R$ and ${\mu }_1,\cdots ,{\mu }_n$ are integers $\ge 0$.

Let $R$ be a left and right Noetherian ring and $C$ a semidualizing $R$-bimodule. We introduce a transpose ${\mathrm{Tr}}_{\mathrm{c}}M$ of an $R$-module $M$ with respect to $C$ which unifies the Auslander transpose and Huang’s transpose, see Z. Y. Huang, On a generalization of the Auslander-Bridger transpose, Comm. Algebra 27 (1999), 5791–5812, in the two-sided Noetherian setting, and use ${\mathrm{Tr}}_{\mathrm{c}}M$ to develop further the generalized Gorenstein dimension with respect to $C$. Especially, we generalize the Auslander-Bridger formula to the generalized...

Let $R$ be a commutative Noetherian ring and $𝔞$ an ideal of $R$. We introduce the concept of $𝔞$-weakly Laskerian $R$-modules, and we show that if $M$ is an $𝔞$-weakly Laskerian $R$-module and $s$ is a non-negative integer such that ${\mathrm{Ext}}_R^j(R/𝔞,H_{𝔞}^i\left(M\right))$ is $𝔞$-weakly Laskerian for all $i<s$ and all $j$, then for any $𝔞$-weakly Laskerian submodule $X$ of $H_{𝔞}^s\left(M\right)$, the $R$-module ${\mathrm{Hom}}_R(R/𝔞,H_{𝔞}^s\left(M\right)/X)$ is $𝔞$-weakly Laskerian. In particular, the set of associated primes of $H_{𝔞}^s\left(M\right)/X$ is finite. As a consequence, it follows that if $M$ is a finitely generated $R$-module and $N$ is an $𝔞$-weakly...

We consider polynomial mappings (f,g) of ℂ² with constant nontrivial jacobian. Using the Riemann-Hurwitz relation we prove among other things the following: If g - c (resp. f - c) has at most two branches at infinity for infinitely many numbers c or if f (resp. g) is proper on the level set $g^{-1}\left(0\right)$ (resp. $f^{-1}\left(0\right)$), then (f,g) is bijective.

2000 Mathematics Subject Classification: Primary 20F55, 13F20; Secondary 14L30.We construct invariant polynomials for the reflection groups [3, 4, 3] and [3, 3, 5] by using some special sets of lines on the quadric P1 × P1 in P3. Then we give a simple proof of the well known fact that the ring of invariants are rationally generated in degree 2,6,8,12 and 2,12,20,30.

In this paper, a new kind of graph on a commutative ring is introduced and investigated. Small intersection graph of a ring $R$, denoted by $G\left(R\right)$, is a graph with all non-small proper ideals of $R$ as vertices and two distinct vertices $I$ and $J$ are adjacent if and only if $I\cap J$ is not small in $R$. In this article, some interrelation between the graph theoretic properties of this graph and some algebraic properties of rings are studied. We investigated the basic properties of the small intersection graph as diameter,...

The purpose of this article is to give, for any (commutative) ring $A$, an explicit minimal set of generators for the ring of multisymmetric functions ${\mathrm{T}S}_A^d\left(A[x_1,\cdots ,x_r]\right)={\left(A{[x_1,\cdots ,x_r]}^{{\otimes }_{A}d}\right)}^{{𝔖}_d}$ as an $A$-algebra. In characteristic zero, i.e. when $A$ is a $\mathbb{Q}$-algebra, a minimal set of generators has been known since the 19th century. A rather small generating set in the general case has also recently been given by Vaccarino but it is not minimal in general. We also give a sharp degree bound on the generators, improving the degree bound previously...