This is the ninth paper by the author dealing with monic or monic free ideals and k-ideals in polynomial semirings S[X], \(X=\{x_ 1,...,x_ n\}\), \(n\geq 1\). All statements need the following assumptions on the semiring \((S,+,\cdot):\) 1) \((S,+)\) and (S,\(\cdot)\) are commutative semigroups with neutral elements 0 and e, respectively. For all a,b\(\in S\), 2) \(a+b=0\Rightarrow a=b=0\), 3) \(0\cdot a=0\) and 4) \(a\cdot b=0\Rightarrow a=0\vee b=0\) are satisfied, where 3) and 4) are not assumed. For each subset \(\emptyset \neq \tau =\{x'_ 1,...,x'_ t\}\subseteq X\), a polynomial \(f\in S[\tau]\) is called saturated if either \(f=0\) or each \(x'_ i\in \tau\) appears in some nonzero term of f. All these polynomials form an ideal \(S_{\tau}\) of S[\(\tau\) ]. For each ideal A of S[X], \(A_{\tau}=A\cap S_{\tau}\) defines an ideal of S[\(\tau\) ], and \(A_{\emptyset}=A\cap S\) one of S. Let \((A_{\tau})\) be the ideal of S[X] generated by \(A_{\tau}\), \(\emptyset \subseteq \tau \subseteq X\), and define the ideals \(P_{\tau}=\sum_{\delta \subseteq \tau}(A_{\delta})\) of S[X]. The main results (Thms. 3.3 and 3.4) are: If the ideal A is monic [a k-ideal], the same holds for all ideals \(P_{\tau}\), \(\tau\subseteq X\), and conversely.
Reviewer’s remarks: a) Because of \(P_ X=A\), it is superfluous to prove the converse direction (which in fact does not hold with the restriction \(\tau\subset X)\). b) The other part of both proofs are based on the statement: ”If \(f+g\) is a saturated polynomial in \(P_{\tau}\), we know that \(f\in A_{\lambda}\) and \(g\in A_{\kappa}\) for some \(\lambda\),\(\kappa\subseteq \tau.''\) But this statement can be disproved by simple counter examples. (The reviewer needed some time to prove both theorems, using completely other ideas.) c) In the example of § 5, for \(A=(2,x^ n+2,y^ m+2,z^ t)\) in \({\mathbb{Z}}^+[X]\) for \(X=\{x,y,z\}\), \(n>m>t>1\), the list of all \(A_{\tau}\) for \(\tau\subseteq X\) is given. All \(A_{\tau}\) are wrong except \(A_{\emptyset}\). E.g. \(A_{\{x,y\}}\) is claimed to be \((x^ n+2)+(y^ m+2)\subseteq {\mathbb{Z}}^+[x,y]\) (apart from 2 misprints). In fact, \(2\cdot h(x,y)\) for all saturated polynomials \(h(x,y)\in {\mathbb{Z}}^+[x,y]\) is also in \(A_{\{x,y\}}\), and all polynomials \((x^ n+2)g(x)\) and \((y^ m+2)g(y)\) are not.