Prime subacts over commutative monoids with zero.
The authors consider centered right \(S\)-acts over a commutative monoid \(S\). For two subacts \(B\) and \(C\) of \(A\), use notation \((B:C)=\{s\in S: Cs\subseteq B\}\). A proper subact \(B\) of \(A\) is called a `prime' (resp. `primary') subact if for every \(s\in S\) and \(a\in A\), \(as\in B\) implies that \(a\in B\) or \(s\in (B:A)\) (resp. \(s\in\mathrm{rad}_S((B:A))\)). Here \(\mathrm{rad}_A(B)\) denotes the intersection of all prime subacts of \(A\) containing \(B\). An \(S\)-act \(A\) is called `prime' if the subact \(\{\theta\}\) of \(A\) is prime. If \(B\) is a prime (resp. primary) subact of \(A\) and \((B:A)=P\) (resp. \(\mathrm{rad}_S((B:A))=P\)) then \(B\) is called `\(P\)-prime' (resp. `\(P\)-primary'). An \(S\)-act \(A\) is said to be `second' (resp. `secondary') if \(A\neq\{\theta\}\) and for all \(s\in S\), either \(As=A\) or \(As=\{\theta\}\) (resp. \(\exists n\in \mathbb N,\;As^n=\{\theta\}\)). Similarly, if \(A\) is a second (resp. secondary) \(S\)-act and \(\mathrm{Ann}(A)=P\) (resp. \(\mathrm{rad}_S(\mathrm{Ann}(A))=P\)) then \(A\) is called `\(P\)-second' (resp. `\(P\)-secondary'). The authors derive a number of basic properties of (\(P\)-)prime, (\(P\)-)primary, (\(P\)-)second and (\(P\)-)secondary (sub)acts.
centered right acts over commutative monoids
scientific article; zbMATH DE number 6115199