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Register a new userOn Topological Lattices and an Application to First Submodules
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We introduce the notion of a (strongly) topological lattice $mathcal{L}=(L,wedge ,vee)$ with respect to a subset $Xsubsetneqq L;$ aprototype is the lattice of (two-sided) ideals of a ring $R,$ which is(strongly) topological with respect to the prime spectrum of $R.$ We investigate and characterize (strongly) topological lattices. Given a non-zero left $R$-module $M,$ we introduce and investigate the spectrum $mathrm{Spec}^{mathrm{f}}(M)$ of textit{first submodules} of $M.$ We topologize $mathrm{Spec}^{mathrm{f}}(M)$ and investigate the algebraic properties of $_{R}M$ by passing to the topological properties of the associated space.
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This paper studies the differential lattice, defined to be a lattice $L$ equipped with a map $d:Lto L$ that satisfies a lattice analog of the Leibniz rule for a derivation. Isomorphic differential lattices are studied and classifications of differential lattices are obtained for some basic lattices. Several families of derivations on a lattice are explicitly constructed, giving realizations of the lattice as lattices of derivations. Derivations on a finite distributive lattice are shown to have a natural structure of lattice. Moreover, derivations on a complete infinitely distributive lattice form a complete lattice. For a general lattice, it is conjectured that its poset of derivations is a lattice that uniquely determines the given lattice.
We investigate ideal-semisimple and congruence-semisimple semirings. We give several new characterizations of such semirings using e-projective and e-injective semimodules. We extend several characterizations of semisimple rings to (not necessarily subtractive) commutative semirings.
Let A denote the ring of differential operators on the affine line with its two usual generators t and d/dt given degrees +1 and -1 respectively. Let X be the stack having coarse moduli space the affine line Spec k[z] and isotropy groups Z/2 at each integer point. Then the category of graded A-modules is equivalent to the category of quasi-coherent sheaves on X. Version 2: corrected typos and deleted appendix at referees suggestion.
The set of all maximal ideals of the ring $mathcal{M}(X,mathcal{A})$ of real valued measurable functions on a measurable space $(X,mathcal{A})$ equipped with the hull-kernel topology is shown to be homeomorphic to the set $hat{X}$ of all ultrafilters of measurable sets on $X$ with the Stone-topology. This yields a complete description of the maximal ideals of $mathcal{M}(X,mathcal{A})$ in terms of the points of $hat{X}$. It is further shown that the structure spaces of all the intermediate subrings of $mathcal{M}(X,mathcal{A})$ containing the bounded measurable functions are one and the same and are compact Hausdorff zero-dimensional spaces. It is observed that when $X$ is a $P$-space, then $C(X) = mathcal{M}(X,mathcal{A})$ where $mathcal{A}$ is the $sigma$-algebra consisting of the zero-sets of $X$.
Let k be a field, q in k. We derive a cup product formula on the Hochschild cohomology ring of a family Lambda_q of quiver algebras. Using this formula, we determine a subalgebra of k[x,y] isomorphic to Hochschild cohomology modulo N, where N is the ideal generated by homogeneous nilpotent elements. We explicitly construct non-nilpotent Hochschild cocycles which cannot be generated by lower homological degree elements, thus disproving the Snashall-Solberg finite generation conjecture.
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