Null ideals of matrices over residue class rings of principal ideal domains

Given a square matrix $A$ with entries in a commutative ring $S$, the ideal of $S[X]$ consisting of polynomials $f$ with $f(A) =0$ is called the null ideal of $A$. Very little is known about null ideals of matrices over general commutative rings. We compute a generating set of the null ideal of a matrix in case $S = D/dD$ is the residue class ring of a principal ideal domain $D$ modulo $d\in D$. We discuss two applications. At first, we compute a decomposition of the $S$-module $S[A]$ into cyclic $S$-modules and explain the strong relationship between this decomposition and the determined generating set of the null ideal of $A$. And finally, we give a rather explicit description of the ring \IntA of all integer-valued polynomials on $A$

Concepts, examples and discussion of influencing factors

Monitoring and surveillance of antimicrobial usage in animals is a public health concern and different methods are currently discussed widely in public, science and politics. The objective of the paper is to present the available methods of monitoring and to discuss possible differences in the assessment of the antibiotics treatment. Sales data are expressed as the average amount of substance per animal or kg live weight (e.g. population‐corrected unit, PCU). The number of Defined Daily Doses (nDDDvet) is calculated by extrapolating sales data with average animal weights and defined drug doses to a number of treatments theoretically applied to animals. In contrast, the number of Used Daily Doses (nUDDvet) displays the actual number of treatments which have been applied. As sales data are relatively easily obtained, they are frequently used. However, their results are influenced by the composition of the population and by the dose of the substances. As both may vary strongly between countries, direct comparison of sales data between countries may be misleading. The concept of analysing sales data is shown by exemplarily using the methods in the ESVAC report 2015. The presentation of usage data in terms of nDDDvet or of nUDDvet increases the comparability of the data from different countries or time periods. Furthermore, fluoroquinolones and third‐/fourth‐generation cephalosporins which, among other substances, bare a potential risk for human health, are used at low doses. Hence, their use contributes to a sales reduction while contrasting the guidelines of prudent use. nDDDvet or nUDDvet have the ability to better reflect the treatment frequency and thus to better link antibiotics use to public health concerns. Quantification of antibiotics should assist to focus on prudent use of antimicrobials to reduce the burden of resistant bacteria and, thus, enhance public health, animal health and animal welfare