let $r$ be a commutative ring with identity and let $m$ be an $r$-module. a proper submodule $p$ of $m$ is called strongly prime submodule if $(p + rx : m)ysubseteq p$ for $x, yin m$, implies that $xin p$ or $yin p$. in this paper, we study more properties of strongly prime submodules. it is shown that a finitely generated $r$-module $m$ is artinian if and only if $m$ is noetherian and every st...

In Bishop-style constructive algebra it is known that if a module over a commutative ring has a Noetherian basis function, then it is Noetherian. Using countable choice we prove the reverse implication for countable and strongly discrete modules. The Hilbert basis theorem for this specific class of Noetherian modules, and polynomials in a single variable, follows with Tennenbaum’s celebrated ve...

Let $X$ be a regular tame stack. If is locally of finite type over field, we prove that the essential dimension equal to its generic dimension, this generalizes previous result P. Brosnan, Z. Reichstein and second author. Now suppose $1$-dimensional local noetherian domain $R$ with fraction field $K$ residue $k$. We $\operatorname{ed}_{k}X_{k} \le \operatorname{ed}_{K}X_{K}$ if $X\to \operatorn...

√(f1,...,fd+1)R[X] [3,p.124]. The question is whether an ideal (f1,...,fd+1) R[X] can be chosen as a reduction of I. We only know the following case of affine domains, which was developed by G. Lyubeznik [4]: Let R be an n-dimensional affine domain over an infinite field k and let I be an ideal of R. Then I has a reduction generated by n+1 elements. He also posed the following conjecture: Let A...

A not necessarily noetherian local ring O is called regular if every finitely generated ideal I O possesses finite projective dimension. In the article localizations O = Aq, q ∈ SpecA, of a finitely presented, flat algebra A over a Prüfer domain R are investigated with respect to regularity: this property of O is shown to be equivalent to the finiteness of the weak homological dimension wdimO. ...

In this expository note, we discuss various aspects of the theory of dimension of schemes, in particular focusing on which hypotheses are necessary in order to insure good behavior, and what sort of pathologies can otherwise arise. References are frequently exercises in disguise (or not in disguise, as the case may be). In particular, because the intended applications are geometric, we will sta...

In this paper we will compare the connectivity dimension c(P/I) of an ideal I in a polynomial ring P with that of any initial ideal of I. Generalizing a theorem of Kalkbrener and Sturmfels [18], we prove that c(P/LT≺(I)) ≥ min{c(P/I), dim(P/I)−1} for each monomial order ≺. As a corollary we have that every initial complex of a Cohen-Macaulay ideal is strongly connected. Our approach is based on...