We call a right module $M$ (strongly) virtually regular if every (finitely generated) cyclic submodule is isomorphic to a direct summand. $M$ is said to be completely virtually regular if every submodule is virtually regular. In this paper, characterizations and some closure properties of the aforementioned modules are given. Several structure results are obtained over commutative rings. In particular, the structures of finitely presented (strongly) virtually regular modules and completely virtually regular modules are fully determined over valuation domains. Namely, for a valuation domain $R$ with the unique maximal ideal $P$, we show that finitely presented (strongly) virtually regular modules are free if and only if $P$ is not principal; and that $P=Rp$ is principal if and only if finitely presented virtually regular modules are of the form $$R^n \oplus (\frac{R}{Rp})^{n_1} \oplus (\frac{R}{Rp^2})^{n_2} \oplus \cdots \oplus (\frac{R}{Rp^k})^{n_k}$$ for nonnegative integers $n,\,k,\,n_1,\,n_2,\cdots ,n_k.$ Similarly, we prove that $P=Rp$ is principal if and only if finitely presented strongly virtually regular modules are of the form $ R^n \oplus (\frac{R}{Rp})^{m}$, where $m,n$ are nonnegative integers.
We also obtain that, $R$ admits a nonzero finitely presented completely virtually regular module $M$ if and only if $P=Rp$ is principal. Moreover, for a finitely presented $R$-module $M$, we prove that: $(i)$ if $R$ is not a DVR, then $M$ is completely virtually regular if and only if $M \cong (\frac{R}{Rp})^{m}$; and $(ii)$ if $R$ is a DVR, then $M$ is completely virtually regular if and only if $M\cong R^n \oplus (\frac{R}{Rp})^{m}.$ Finally, we obtain a characterization of finitely generated virtually regular modules over the ring of integers.
一个$R$ -模块$M$被称为实际上是不对称的，如果对于每个不对称生成的子模块$0 \neq K \subseteq M$，$K/$ Rad $(K)$实际上是简单的。本文通过去掉虚单条件而代之以循环一致条件，推广了虚单列模。一个$R$ -模块$M$被称为循环均匀unissimplex，如果$K/$ Rad $(K)$是循环的和均匀的，对于每个循环生成的子模块$0 \neq K \subseteq M$。另外，如果$M$是循环均匀单列模的直和，则称它是循环均匀串行的。给出了循环一致（单）序列模和环的几个性质。进一步刻画了Noether左循环一致均匀环的结构。最后，我们研究了环$R$具有的性质：每个环生成的$R$ -模是循环一致串行的。
Note on length and multiplicity of modules over two-dimensional regular local rings
Okayama University（冈山大学）；The Institute of Mathematical Sciences, Chennai, India（印度金奈数学科学研究所）；Homi Bhabha National Institute, Mumbai, India（印度孟买霍米·巴巴国立研究所）
Hilbert regularity of ZZ-graded modules over polynomial rings
设M是标准分次多项式环R = K [X_1，…，X_n]，K为域，设H_M（t）= Q_M（t）/（1-t）^d为M的Hilbert级数.我们引入M的Hilbert正则性作为具有Hilbert级数H_M的R-模的Castelnuovo-Mumford正则性的最低可能值。我们的主要结果是这个不变量的一个算术描述，它将M的Hilbert正则性与最小k联系起来，使得幂级数Q_M（1-t）/（1-t）^k没有负系数.最后给出了R-模的Hilbert正则性和Hilbert深度的计算算法。
Some criteria for regular and Gorenstein local rings via syzygy modules
设$ R $是Cohen-Macaulay局部环.证明了极大Cohen-Macaulay $ R $ -模的$ n $次合模不可能对每个$ n \ge 1 $都有半对偶直和项.特别地，可以得出$ R $是Gorenstein当且仅当$ R $的典范模的某个合点有非零自由直和项。给出了最小重数的Cohen-Macaulay局部环是正则环或Gorenstein环的若干充要条件。这些标准是基于消失的某些Ext或TOR涉及合合模的剩余域。
