Let $M$ be a free module over a noetherian ring. For ${\omega }_1,...,{\omega }_k\in M$, let $𝒜$ be the ideal generated by coefficients of ${\omega }_1\wedge ...\wedge {\omega }_k$. For an element $\omega \in {\bigwedge }^{p}M$ with $p\<\mathrm{prof}.𝒜$, if $\omega \wedge {\omega }_1\wedge ...\wedge {\omega }_k=0$, there exists ${\eta }_1,...,{\eta }_k\in {\bigwedge }^{p-1}M$ such that $\omega ={\sum }_{i=1}^k{\eta }_i\wedge {\omega }_i$.This is a generalization of a lemma on the division of forms due to de Rham (Comment. Math. Helv., 28 (1954)) and has some applications to the study of singularities.

Let $D$ be an integral domain with the quotient field $K$, $X$ an indeterminate over $K$ and $x$ an element of $D$. The Bhargava ring over $D$ at $x$ is defined to be ${𝔹}_x\left(D\right):=\{f\in K\left[X\right]:\text{for}\text{all}a\in D,f(xX+a)\in D\left[X\right]\}$. In fact, ${𝔹}_x\left(D\right)$ is a subring of the ring of integer-valued polynomials over $D$. In this paper, we aim to investigate the behavior of ${𝔹}_x\left(D\right)$ under localization. In particular, we prove that ${𝔹}_x\left(D\right)$ behaves well under localization at prime ideals of $D$, when $D$ is a locally finite intersection of localizations. We also attempt a classification of integral domains $D$...

The dual of a Gorenstein module is called a co-Gorenstein module, defined by Lingguang Li. In this paper, we prove that if R is a local U-ring and M is an Artinian R-module, then M is a co-Gorenstein R-module if and only if the complex $Ho{m}_{R̂}((,R̂),M)$ is a minimal flat resolution for M when we choose a suitable triangular subset on R̂. Moreover we characterize the co-Gorenstein modules over a local U-ring and Cohen-Macaulay local U-ring.

In this paper, we use a characterization of $R$-modules $N$ such that $f{d}_{R}N=p{d}_{R}N$ to characterize Cohen-Macaulay rings in terms of various dimensions. This is done by setting $N$ to be the $dth$ local cohomology functor of $R$ with respect to the maximal ideal where $d$ is the Krull dimension of $R$.

We prove that for a commutative ring $R$, every noetherian (artinian) $R$-module is quasi-injective if and only if every noetherian (artinian) $R$-module is quasi-projective if and only if the class of noetherian (artinian) $R$-modules is socle-fine if and only if the class of noetherian (artinian) $R$-modules is radical-fine if and only if every maximal ideal of $R$ is idempotent.

In this paper we study commutative rings $R$ whose prime ideals are direct sums of cyclic modules. In the case $R$ is a finite direct product of commutative local rings, the structure of such rings is completely described. In particular, it is shown that for a local ring $(R,ℳ)$, the following statements are equivalent: (1) Every prime ideal of $R$ is a direct sum of cyclic $R$-modules; (2) $ℳ={⨁}_{\lambda \in \Lambda }R{w}_{\lambda }$ where $\Lambda$ is an index set and $R/Ann\left(w_{\lambda }\right)$ is a principal ideal ring for each $\lambda \in \Lambda$; (3) Every prime ideal of $R$ is a direct sum of at most...

If $A$ is a domain with the ascending chain condition on (integral) invertible ideals, then the group $I(A)$ of its invertible ideals is generated by the set $I_m(A)$ of maximal invertible ideals. In this note we study some properties of $I_m(A)$ and we prove that, if $I(A)$ is a free group on $I_m(A)$, then $A$ is a locally factorial Krull domain.

Let $R$ be a ring with an identity (not necessarily commutative) and let $M$ be a left $R$-module. This paper deals with multiplication and comultiplication left $R$-modules $M$ having right ${End}_R\left(M\right)$-module structures.