Multiplicative ideal theory in the context of commutative monoids

Studying monoids is not enough to study multiplicative properties of rings: an elementary approach

“…Our coverage of this topic will be tool specific, in that we shall concentrate more on what we need to establish our thesis. For this the best source we find in the field is Franz Halter-Koch's book [10] where, between the lines, further evidence to the aim of the present paper can be found.…”

“…Our coverage of this topic will be tool specific, in that we shall concentrate more on what we need to establish our thesis. For this the best source we find in the field is Franz Halter-Koch's book [10] where, between the lines, further evidence to the aim of the present paper can be found.Before we start the review, we wish to point out that the material presented here is a sort of fact-presenting mission and it is not our intention to belittle monoids which have their own uses and play active roles in various contexts. Moreover, on the other side, it is known that monoid theory can produce results of ring-theoretical relevance which cannot be derived from the ring structure alone.…”

“…Moreover, on the other side, it is known that monoid theory can produce results of ring-theoretical relevance which cannot be derived from the ring structure alone. For example, in a Noetherian domain, the monoid of elements coprime to the conductor forms a Krull monoid, whose arithmetic properties cannot be detected by purely ring-theoretical methods [10, Chapter 22]. …”

Onv-Domains and Star Operations

“…Given an ideal system r, r-Dedekind monoids are introduced and studied in [23,Chapter 23,§3]. However, the notion of * -Dedekind domain coincides with that in [23] in case * = * f or in the case of Krull domains, but they are different in general (e.g., v-Dedekind domains are precisely Krull domains, but v-Dedekind monoids are just completely integrally closed monoids).…”

“…Therefore a P * MD is a * -Prüfer domain, but the converse is not true since there are v-domains (= v-Prüfer domains) that are not PvMD's [26]. Also note that for an ideal system r on a monoid, the notion of r-Prufer monoid, for a general r, was introduced in [23,Chapter 17]. However, most of the results on r-Prufer monoids in [23] were proved for r-finitary.…”