Behaviour of krull dimension under extension of the base field

1991 ◽  
Vol 19 (1) ◽  
pp. 143-156
Author(s):  
Timothy J. Hodges ◽  
Kyunghee Kim ◽  
Richard Resco
Keyword(s):  
Krull Dimension ◽  
Base Field

scholarly journals BOUNDED AND FULLY BOUNDED MODULES

2011 ◽  
Vol 84 (3) ◽  
pp. 433-440
Author(s):  
A. HAGHANY ◽  
M. MAZROOEI ◽  
M. R. VEDADI
Keyword(s):  
Krull Dimension ◽  
Finitely Generated ◽  
Morita Invariant ◽  
Prime Submodule ◽  
Invariant Properties ◽  
Noetherian Modules

AbstractGeneralizing the concept of right bounded rings, a module MR is called bounded if annR(M/N)≤eRR for all N≤eMR. The module MR is called fully bounded if (M/P) is bounded as a module over R/annR(M/P) for any ℒ2-prime submodule P◃MR. Boundedness and right boundedness are Morita invariant properties. Rings with all modules (fully) bounded are characterized, and it is proved that a ring R is right Artinian if and only if RR has Krull dimension, all R-modules are fully bounded and ideals of R are finitely generated as right ideals. For certain fully bounded ℒ2-Noetherian modules MR, it is shown that the Krull dimension of MR is at most equal to the classical Krull dimension of R when both dimensions exist.

Regular Hom-algebras admitting a multiplicative basis

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Antonio J. Calderón Martín
Keyword(s):  
Arbitrary Dimension ◽  
Base Field ◽  
Direct Sum ◽  
Multiplicative Basis ◽  
Arbitrary Base ◽  
The Family

AbstractLet {({\mathfrak{H}},\mu,\alpha)} be a regular Hom-algebra of arbitrary dimension and over an arbitrary base field {{\mathbb{F}}}. A basis {{\mathcal{B}}=\{e_{i}\}_{i\in I}} of {{\mathfrak{H}}} is called multiplicative if for any {i,j\in I}, we have that {\mu(e_{i},e_{j})\in{\mathbb{F}}e_{k}} and {\alpha(e_{i})\in{\mathbb{F}}e_{p}} for some {k,p\in I}. We show that if {{\mathfrak{H}}} admits a multiplicative basis, then it decomposes as the direct sum {{\mathfrak{H}}=\bigoplus_{r}{{\mathfrak{I}}}_{r}} of well-described ideals admitting each one a multiplicative basis. Also, the minimality of {{\mathfrak{H}}} is characterized in terms of the multiplicative basis and it is shown that, in case {{\mathcal{B}}}, in addition, it is a basis of division, then the above direct sum is composed by means of the family of its minimal ideals, each one admitting a multiplicative basis of division.

Krull dimension of permutation modules

1993 ◽  
Vol 21 (2) ◽  
pp. 705-710
Author(s):  
Robert L. Snider
Keyword(s):  
Krull Dimension

Global dimension of rings with krull dimension

1992 ◽  
Vol 20 (10) ◽  
pp. 2863-2876 ◽  
Author(s):  
John J. Koker
Keyword(s):  
Global Dimension ◽  
Krull Dimension

scholarly journals Differential Modules on p-Adic Polyannuli—Erratum

2010 ◽  
Vol 9 (3) ◽  
pp. 669-671 ◽  
Author(s):  
Kiran S. Kedlaya ◽  
Liang Xiao
Keyword(s):  
Base Field ◽  
Convergence Theorems ◽  
Differential Modules

The statement of Theorem 2.7.6 in the indicated paper is incorrect. For instance, if m = 0 (i.e., the base field K contains no additional derivations), it is inconsistent with Theorem 2.7.4 due to the distinction between intrinsic and extrinsic generic radii of convergence. Theorems 2.7.12 and 2.7.13 are incorrect for similar reasons.

scholarly journals FRAMED CORRESPONDENCES AND THE MILNOR–WITT -THEORY

2016 ◽  
Vol 17 (4) ◽  
pp. 823-852 ◽  
Author(s):  
Alexander Neshitov
Keyword(s):  
Characteristic Zero ◽  
Triangulated Category ◽  
Algebraic Varieties ◽  
Base Field ◽  
Witt Theory

Given a field $k$ of characteristic zero and $n\geqslant 0$, we prove that $H_{0}(\mathbb{Z}F(\unicode[STIX]{x1D6E5}_{k}^{\bullet },\mathbb{G}_{m}^{\wedge n}))=K_{n}^{MW}(k)$, where $\mathbb{Z}F_{\ast }(k)$ is the category of linear framed correspondences of algebraic $k$-varieties, introduced by Garkusha and Panin [The triangulated category of linear framed motives $DM_{fr}^{eff}(k)$, in preparation] (see [Garkusha and Panin, Framed motives of algebraic varieties (after V. Voevodsky), Preprint, 2014, arXiv:1409.4372], § 7 as well), and $K_{\ast }^{MW}(k)$ is the Milnor–Witt $K$-theory of the base field $k$.

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