The concept of pseudo valuations on hoop-algebras is introduced, and then the relationship between pseudo-valuations filers investigated. Some conditions for a real-valued function to be valuation are provided. Finally, we prove fact that binary operations uniformly continuous based notion valuation.

The ideals of the ring Zp are {0} and pZp, n ≥ 0. From this it follows that Zp is a discrete valuation ring, a principal ideal domain with exactly one maximal ideal, namely pZp; Zp is the valuation ring of Qp with the valuation vp. For n ≥ 1, Zp/pZp is isomorphic as a ring with Z/pZ. |x|p = p−vp(x), dp(x, y) = |x− y|p. With the topology induced by the metric dp, Qp is a locally compact abelian ...

We prove that if $R$ is a commutative Noetherian ring, then every countably generated flat $R$-module quite flat, i.e., direct summand of transfinite extension localizations in countable multiplicative subsets. also show the spectrum cardinality less than $\kappa$, where $\kappa$ an uncountable regular cardinal, modules with generators. This provides alternative proof fact over ring spectrum, a...

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...

The five-membered ring of the title compound, C(10)H(14)NO, is almost planar [mean deviation from best plane = 0.006 (1) Å]. The N-O bond is in the plane of the five-membered ring. The mol-ecule is positioned about a pseudo-mirror plane at y = 0.375. In the crystal, mol-ecules are connected by inter-molecular C-H⋯O contacts into layers parallel to (010).