Messages in communication networks often are considered as "discrete" taking values in some finite alphabet (e.g. a finite field). However, if we want to consider for example communication based on analogue signals, we will have to consider messages that might be functions selected from an infinite function space. In this paper, we extend linear network coding over finite/discrete alphabets/message space to the infinite/continuous case. The key to our approach is to view the space of operators that acts linearly on a space of signals as a module over a ring.
It turns out that modules over many rings $R$ leads to unrealistic network models where communication channels have unlimited capacity. We show that a natural condition to avoid this is equivalent to the ring $R$ being Dedekind finite (or Neumann finite) i.e. each element in $R$ has a left inverse if and only if it has a right inverse. We then consider a strengthened capacity condition and show that this requirement precisely corresponds to the class of (faithful) modules over stably finite rings (or weakly finite).
The introduced framework makes it possible to compare the performance of digital and analogue techniques. It turns out that within our model, digital and analogue communication outperforms each other in different situations. More specifically we construct: 1) A communications network where digital communication outperforms analogue communication. 2) A communication network where analogue communication outperforms digital communication.
The performance of a communication network is in the finite case usually measured in terms band width (or capacity). We show this notion also remains valid for finite dimensional matrix rings which make it possible (in principle) to establish gain of digital versus analogue (analogue versus digital) communications.
Hereditarily and Super Bassian Modules over Certain Rings
Bulgarian Academy of Sciences, Institute of Mathematics and Informatics（保加利亚科学院数学与信息学研究所）；Faculty of Mathematics, The University of Danang - University of Science and Education（岘港大学-科学教育大学数学学院）；Department of Algebra, MFF UK（查理大学数学与物理学院代数系）
Determination of the stably free cancellation property for orders
Ludwig-Maximilians-Universität München（慕尼黑路德维希-马克西米利安大学）；Naturwissenschaftlich-Technische Fakultät Universität Siegen（锡根大学自然科学技术学院）；University of Exeter（埃克塞特大学）
Content Algebras Over Commutative Rings With Zero-Divisors
设$M$是$R$ -模，$c$是从$M$到由$c(x) = \cap \lbrace I \colon I \text{is an ideal of} R \text{and} x \in IM \rbrace $定义的$R$的理想的函数.对于所有$x \in M$，如果是$x \in c(x)M $，则$M$被称为内容$R$模块。$B$称为容$R$ -代数，如果它是一个忠实平坦的容$R$ -模并且满足Dedekind-Mertens容度公式.本文利用理想理论方法证明了容度模和代数的一些新结果。
