Let H be an integral domain, and let Σ be a collection of integrally closed overrings of H. We show that if A is an overring of H such that H = ( T R∈Σ R)∩A, and if Σ is a Noetherian subspace of the space of all integrally closed overrings of H, then there exists a weakly Noetherian subspace Γ of integrally closed overrings of H such that H = ( T R∈Γ R) ∩ A, and no member of Γ can be omitted fr...

In this paper we study the log structures on generalized semistable varieties, generalize the result by F. Kato and M. Olsson, and prove the canonicity of log structure when it can be expected. In out text we first give the definitions of local chart and weakly normal crossing morphism. Then we study the invariants of complete noetherian local ring coming from weakly normal crossing morphisms. ...

Let $R$ be a commutative ring with identity‎. ‎A proper ideal $P$ of $R$ is a $(n-1,n)$-$Phi_m$-prime ($(n-1,n)$-weakly prime) ideal if $a_1,ldots,a_nin R$‎, ‎$a_1cdots a_nin Pbackslash P^m$ ($a_1cdots a_nin Pbackslash {0}$) implies $a_1cdots a_{i-1}a_{i+1}cdots a_nin P$‎, ‎for some $iin{1,ldots,n}$; ($m,ngeq 2$)‎. ‎In this paper several results concerning $(n-1,n)$-$Phi_m$-prime and $(n-1,n)$-...

let $r$ be a commutative ring with identity‎. ‎a proper ideal $p$ of $r$ is a $(n-1,n)$-$phi_m$-prime ($(n-1,n)$-weakly prime) ideal if $a_1,ldots,a_nin r$‎, ‎$a_1cdots a_nin pbackslash p^m$ ($a_1cdots a_nin pbackslash {0}$) implies $a_1cdots a_{i-1}a_{i+1}cdots a_nin p$‎, ‎for some $iin{1,ldots,n}$; ($m,ngeq 2$)‎. ‎in this paper several results concerning $(n-1,n)$-$phi_m$-prime and $(n-1,n)$-...

A compact LOTS has an ω 1 -like base if and only if it is metrizable. An infinite cardinal κ is the Noetherian type of a compact LOTS if and only if κ 6= ω1 and κ is not weakly inaccessible. If X is a homogeneous compactum with regular weight, then every base of X contains an Nt(X)op-like

We associate to every equicharacteristic zero Noetherian local ring R a faithfully flat ring extension which is an ultraproduct of rings of various prime characteristics, in a weakly functorial way. Since such ultraproducts carry naturally a non-standard Frobenius, we can define a new tight closure operation on R by mimicking the positive characteristic functional definition of tight closure. T...

Since the theory of ideals plays an important role in the theory of quotient semirings, in this paper, we will make an intensive study of the notions of Noetherian, Artinian, prime, primary, weakly primary and k-maximal ideals in commutative quotient semirings. The bulk of this paper is devoted to stating and proving analogues to several well-known theorems in the theory of the residue class ri...

An overview is given on the decidability results that have been obtained for the various classes of nite monoid-presentations that involve string-rewriting systems which are noetherian and (weakly) connuent. Further, the Knuth-Bendix completion procedure and some of its extensions are described. Finally, a current research project supported by the Deutsche Forschungsgemeinschaft (DFG) is presen...

We study the algebraic and arithmetic structure of monoids invertible ideals (more precisely, $r$-invertible $r$-ideals for certain ideal systems $r$) Krull weakly Mori domains. also investigate all nonzero polynomial rings with at least two indeterminates over noetherian Among others, we show that they are not transfer but share several arithmetical phenomena having infinite class group prime ...

In the theory of modules over commutative rings there are several possibilities of defining associated prime ideals. The usual definition of an associated prime ideal p for a module M is that p is the annihilator of an element of M . In [2] §1 exercise 17 a generalization of this notion is given. p is called weakly associated (faiblement associé) to M if p is minimal in the set of the prime ide...