On the groups of units of finite commutative chain rings. (Q1867477)

A commutative chain ring is a commutative ring whose ideals form a chain under set theoretic inclusion. Let \(M\) be the maximal ideal of a finite commutative chain ring \(R\) with characteristic \(p^n\), then \(R/M\) is a finite field, say \(R/M \cong \text{GF}(p^r)\), and \(pR=M^e\), \(e \leq s\), where \(s\) is the nilpotency of \(M\). When \(p-1\) is not a divisor of \(e\), the structure of the group of units of \(R\) is known (which depends only on the parameters \(p,n,r,e,s\)). In the paper, the authors give an algorithm to compute the structure of the group of units when \(p-1\) is a divisor of \(e\). Such a structure depends on the parameters \(p,n,r,e,s\), and the Eisenstein polynomial that defines \(R\) as an extension over the Galois ring \(\text{GR}(p^n,r)\). The authors also refine the known result for the case \((p-1) \nmid e\) by listing a set of linearly independent generators for the group of units of \(R\).