Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0906.0282

Real closed * reduced partially ordered rings

Jose Capco
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this work we attempt to generalize our result in [6] [7] for real rings (not just von Neumann regular real rings). In other words we attempt to characterize and construct real closure * of commutative unitary rings that are real. We also make some very interesting and significant discoveries regarding maximal partial orderings of rings, Baer rings and essentail extension of rings. The first Theorem itself gives us a noteworthy bijection between maximal partial orderings of two rings by which one is a rational extension of the other. We characterize conditions when a Baer reduced ring can be integrally closed in its total quotient ring. We prove that Baer hulls of rings have exactly one automorphism (the identity) and we even prove this for a general case (Lemma 12). Proposition 14 allows us to study essential extensions of rings and their relation with minimal prime spectrum of the lower ring. And Theorem 15 gives us a construction of the real spectrum of a ring generated by adjoining idempotents to a reduced commutative subring (for instance the construction of Baer hull of reduced commutative rings). From most of the above interesting theories we prove that there is a bijection between the real closure * of real rings (upto isomorphisms) and their maximal partial orderings. We then attempt to develop some topological theories for the set of real closure * of real rings (upto isomorphism) that will help us give a topological characterization in terms of the real and prime spectra of these rings. The topological characterization will be revealed in a later work. It is noteworthy to point out that we can allow ourself to consider mostly the minimal prime spectrum of the real ring in order to develop our topological theories.

Explore related subjects

math.RAmath.ACmath.AG

Keep this discovery

BibTeXRIS

Jose Capco. 2009-06-01. Real closed * reduced partially ordered rings. https://arxiv.org/abs/0906.0282

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Invariants of Nilpotent Lie Algebras via Geometry and Algebra with a Focus on Computation

We consider the problem of computing rational invariants of nilpotent Lie algebras. We compare two methods that are commonly used for this task: the method of integral curves and the Dixmier map. Given a derivation of a rational function field with polynomial coefficients, we formulate a condition under which the kernel can be recovered from a family of rational integral curves, and we show that triangular derivations satisfy this hypothesis. This yields an explicit description of the kernel as a purely transcendental extension and produces algebraically independent generators. We also show that, in the triangular case, the resulting generators agree with those obtained from the Dixmier map via a local slice. A careful analysis of the generating set obtained from this method leads to an algorithm for computing generators of the rational invariant field of a nilpotent Lie algebra. An implementation of the methods is available in the SageMath system.

math.RA↗
arXiv2026

Quasilinear multiplication in the real Cayley--Dickson tower

Direct evaluation of the defining product in the real Cayley--Dickson algebra $A_n$, of dimension $N=2^n$, has quadratic arithmetic complexity. This paper gives a uniform algorithm for multiplication using $O(N\log N)$ real arithmetic operations and $O(N)$ auxiliary storage. The algorithm reduces multiplication to the alternating product on the imaginary subspace, then evaluates that product by a two-call recursion over one fixed quadratic coefficient extension. For $n\ge1$, the resulting bilinear algorithm uses at most $(9n-15)2^{n-1}+10$ input-dependent real multiplications, and for $n\ge3$, the specified arithmetic schedule uses $(34n-83)2^{n-1}+50$ real operations in total. Under this counting convention, the quasilinear schedule uses fewer operations than direct multiplication for $N\ge16$ and than the uniform Cariow--Cariowa method for $N\ge32$. The algorithm is implemented in the MIT-licensed C11 library fastCD, with a NumPy-backed Python interface, and its results are checked against an independent implementation of the defining recursion. In single-core benchmarks against direct multiplication and the uniform Cariow--Cariowa method, the quasilinear implementation had the lowest mean time of the three at every tested dimension $N\ge32$, for both single and batched products, and was roughly $16$ times faster than direct multiplication at $N=1024$.

math.RA↗
arXiv2026

Graded classification of Leavitt path algebras in terms of strong shift equivalence

Given two finite essential adjacency matrices $A$ and $B$, Hazrat's graded classification conjectures posit that an order preserving $\mathbb{Z}[x,x^{-1}]$-module isomorphism of $K_0$ groups implies graded Morita equivalence of the Leavitt path algebras of $A$ and $B$, while the pointed version predicts a graded isomorphism of the Leavitt path algebras when the $K_0$ group isomorphism additionally preserves the class of the regular module. For any field $k$, we show that the Leavitt path algebras over $k$ of $A$ and $B$ are graded Morita equivalent if and only if $A$ and $B$ are strong shift equivalent. By appealing to counterexamples of Kim and Roush from symbolic dynamics, this shows that Hazrat's graded classification conjectures are false.

math.RA↗