### A structure sheaf on the projective spectrum of a graded fully bounded Noetherian ring.

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In this paper we improve recent results dealing with cellular covers of R-modules. Cellular covers (sometimes called colocalizations) come up in the context of homotopical localization of topological spaces. They are related to idempotent cotriples, idempotent comonads or coreflectors in category theory. Recall that a homomorphism of R-modules π: G → H is called a cellular cover over H if π induces an isomorphism $\pi \u204e:Ho{m}_{R}(G,G)\cong Ho{m}_{R}(G,H)$, where π⁎(φ) = πφ for each $\phi \in Ho{m}_{R}(G,G)$ (where maps are acting on the left). On the one hand,...

This is a summary of some of the main results in the monograph Faithfully Ordered Rings (Mem. Amer. Math. Soc. 2015), presented by the first author at the ALANT conference, Będlewo, Poland, June 8-13, 2014. The notions involved and the results are stated in detail, the techniques employed briefly outlined, but proofs are omitted. We focus on those aspects of the cited monograph concerning (diagonal) quadratic forms over preordered rings.

If $X$ is a Tychonoff space, $C\left(X\right)$ its ring of real-valued continuous functions. In this paper, we study non-essential ideals in $C\left(X\right)$. Let $a$ be a infinite cardinal, then $X$ is called $a$-Kasch (resp. $\overline{a}$-Kasch) space if given any ideal (resp. $z$-ideal) $I$ with $gen\phantom{\rule{0.166667em}{0ex}}\left(I\right)<a$ then $I$ is a non-essential ideal. We show that $X$ is an ${\aleph}_{0}$-Kasch space if and only if $X$ is an almost $P$-space and $X$ is an ${\aleph}_{1}$-Kasch space if and only if $X$ is a pseudocompact and almost $P$-space. Let ${C}_{F}\left(X\right)$ denote the socle of $C\left(X\right)$. For a topological space $X$ with only...

In this paper we introduce the notion of the structure space of $\Gamma $-semigroups formed by the class of uniformly strongly prime ideals. We also study separation axioms and compactness property in this structure space.

Left selfdistributive rings (i.e., $xyz=xyxz$) which are semidirect sums of boolean rings and rings nilpotent of index at most 3 are studied.

Using quantum sections of filtered rings and the associated Rees rings one can lift the scheme structure on Proj of the associated graded ring to the Proj of the Rees ring. The algebras of interest here are positively filtered rings having a non-commutative regular quadratic algebra for the associated graded ring; these are the so-called gauge algebras obtaining their name from special examples appearing in E. Witten's gauge theories. The paper surveys basic definitions and properties but concentrates...

Like the classical Gram-Schmidt theorem for symplectic vector spaces, the sheaf-theoretic version (in which the coefficient algebra sheaf $\mathcal{A}$ is appropriately chosen) shows that symplectic $\mathcal{A}$-morphisms on free $\mathcal{A}$-modules of finite rank, defined on a topological space $X$, induce canonical bases (Theorem 1.1), called symplectic bases. Moreover (Theorem 2.1), if $(\mathcal{E},\phi )$ is an $\mathcal{A}$-module (with respect to a $\u2102$-algebra sheaf $\mathcal{A}$ without zero divisors) equipped with an orthosymmetric $\mathcal{A}$-morphism, we show, like in the classical...