The valuation monoids and pseudo-valuation monoids have been established through valuation domains and pseudo-valuation domains respectively. In this study we continue these lines to describe the almost valuation monoids, almost pseudo-valuation monoids and pseudoalmost valuation monoids. Further we also characterized the newly described monoids as the spirit of valuation monoids pseudo-valuati...

In this study, we generalize the comparability conditions, addressed in Comparability of ideals and valuation over rings [2], between certain maximal ideals and fractional ideals of D which also force D to be a quasi-local domain. Also, we introduce the notion of an almost totally ordered group and establish that: “An integral domain D is an AVD if and only if the group of divisibility of D is ...

Let R be a local ring of bounded module type. It is shown that R is an almost maximal valuation ring if there exists a non-maximal prime ideal J such that R/J is an almost maximal valuation domain. We deduce from this that R is almost maximal if one of the following conditions is satisfied: R is a Q-algebra of Krull dimension ≤ 1 or the maximal ideal of R is the union of all non-maximal prime i...

It is proved that if R is a valuation domain with maximal ideal P and if RL is countably generated for each prime ideal L, then R R is separable if and only RJ is maximal, where J = ∩n∈NP . When R is a valuation domain satisfying one of the following two conditions: (1) R is almost maximal and its quotient field Q is countably generated (2) R is archimedean Franzen proved in [2] that R is separ...

Let R be a pseudo-valuation domain with associated valuation domain V and I a nonzero proper ideal of R. Let R̂ (resp., V̂ ) be the I-adic (resp., IV -adic) completion of R (resp., V ). We show that R̂ is a pseudo-valuation domain (which may be a field); and that if I 6= I2, then V̂ is the associated valuation domain of R̂. Let R be an SFT globalized pseudo-valuation domain with associated Prüfer do...

Let R be a commutative integral domain with quotient field K and let P be a nonzero strongly prime ideal of R. We give several characterizations of such ideals. It is shown that (P : P) is a valuation domain with the unique maximal ideal P. We also study when P^{&minus1} is a ring. In fact, it is proved that P^{&minus1} = (P : P) if and only if P is not invertible. Furthermore, if P is invertib...

It is proven that each indecomposable injective module over a valuation domain R is polyserial if and only if each maximal immediate extension R̂ of R is of finite rank over the completion R̃ of R in the R-topology. In this case, for each indecomposable injective module E, the following invariants are finite and equal: its Malcev rank, its Fleischer rank and its dual Goldie dimension. Similar res...