We give a simple geometric proof of Mohan Kumar's result about complete intersections.

We show how to use the extension and torsion functors in order to compute the torsion submodule of a differential module associated with a multidimensional control system. In particular, we show that the concept of the weak primeness of matrices corresponds to the torsion-freeness of a certain module.

Sullivan associated a uniquely determined $DGA{|}_{\mathbf{Q}}$ to any simply connected simplicial complex $E$. This algebra (called minimal model) contains the total (and exactly) rational homotopy information of the space $E$. In case $E$ is the total space of a principal $G$-bundle, ($G$ is a compact connected Lie-group) we associate a $G$-equivariant model $U_G\left[E\right]$, which is a collection of “$G$-homotopic” $DGA$’s${|}_{\mathbf{R}}$ with $G$-action. $U_G\left[E\right]$ will, in general, be different from the Sullivan’s minimal model of the space $E$. $U_G\left[E\right]$ contains the total rational...

There is a classical result known as Baer’s Lemma that states that an $R$-module $E$ is injective if it is injective for $R$. This means that if a map from a submodule of $R$, that is, from a left ideal $L$ of $R$ to $E$ can always be extended to $R$, then a map to $E$ from a submodule $A$ of any $R$-module $B$ can be extended to $B$; in other words, $E$ is injective. In this paper, we generalize this result to the category $q_{\omega }$ consisting of the representations of an infinite line quiver. This generalization of Baer’s Lemma...

We introduce a class of rings which is a generalization of reflexive rings and $J$-reversible rings. Let $R$ be a ring with identity and $J\left(R\right)$ denote the Jacobson radical of $R$. A ring $R$ is called $J$-reflexive if for any $a,b\in R$, $aRb=0$ implies $bRa\subseteq J\left(R\right)$. We give some characterizations of a $J$-reflexive ring. We prove that some results of reflexive rings can be extended to $J$-reflexive rings for this general setting. We conclude some relations between $J$-reflexive rings and some related rings. We investigate some extensions of...

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.