arXiv subjects

R. Raphael

Publications and source records attributed to R. Raphael.

6 recordsLinked to original sources

arXiv2026

Adjoining Idempotents to a Commutative Ring preprint version

Everything takes place in the category of commutative unitary rings. For a fixed ring $R$, $\alg{R}$ is the class of $R$-algebras and $\igr{R}$ the subclass of idempotent generated $R$-algebras. Following Bezhanishvili et al and their study of Specker and locally Specker $R$-algebras, this paper studies the interplay of properties of $R$ and $A\in \igr{R}$ (both as rings and as $R$-modules). Examples: (1) If $R\sbq A\in \igr{R}$ and $R$ is weak Baer (aka p.p.\ ring) and $A$ is ring essential over $R$, then $A$ is weak Baer and locally Specker. (2) If $R$ is semiprime and all the idempotents of the complete ring of quotients are adjoined to $R$ to form $A$, then $A_R$ is flat iff $R$ is weak Baer, in which case $A$ is locally Specker. The Pierce sheaf is often used since it is based on idempotents. Properties are examined, old and new, that are true for $R$ iff they are true for all the Pierce stalks. Among the new is the result for f-rings (pure ideals are generated by idempotents): $R$ is an f-ring iff each of its Pierce stalks has no non-trivial pure ideals. This allows the expansion of the known classes of f-rings; f-rings play important roles in $\igr{R}$.

math.AC↗

arXiv2022

On extending C^k functions from an open set to R, with applications

For $k\in \N\cup \{\infty\}$ and $U$ open in $ \R$, let $\C^{\,k}(U)$ be the ring of real valued functions on $U$ with the first $k$ derivatives continuous. It is shown for $f\in \C^{\,k}(U)$ there is $g\in \ck{\infty}$ with $U\sbq \coz g$ and $h\in \ck{k}$ with $fg\res_U=h\res_U$. The function $f$ and its $k$ derivatives are not assumed to be bounded on $U$. The function $g$ is constructed using splines based on the Mollifier function. Some consequences about the ring $\ck{k}$ are deduced from this, in particular that $\qcl(\ck{k}) = \text{Q}(\ck{k})$.

math.AC↗

arXiv2020

The reduced ring order and lower semi-lattices II

This paper continues the study of the reduced ring order (rr-order) in reduced rings where $a< b$ if $a^2= ab$. A reduced ring is called rr-good if it is a lower semi-lattice in the order. Examples include weakly Baer rings (wB or PP-rings) but many more. Localizations are examined relating to this order as well as the Pierce sheaf. Liftings of rr-orthogonal sets over surjections of reduced rings are studied. A known result about commutative powers series rings over wB rings is extended, via methods developed here, to very general, not necessarily commutative, power series rings defined by an ordered monoid, showing that they are wB.

math.RA↗

arXiv2020

Tychonoff spaces and a ring theoretic order on $\text{C}(X)$

The reduced ring order (rr-order) is a natural partial order on a reduced ring $R$ given by $r\le_{\text{rr}} s$ if $r^2=rs$. It can be studied algebraically or topologically in rings of the form $\text{C}(X)$. The focus here is on those reduced rings in which each pair of elements has an infimum in the rr-order, and what this implies for $X$. A space $X$ is called rr-good if $\text{C}(X)$ has this property. Surprisingly both locally connected and basically disconnected spaces share this property. The rr-good property is studied under various topological conditions including its behaviour under Cartesian products. The product of two rr-good spaces can fail to be rr-good (e.g., $\beta \mathbf{R}\times \beta \mathbf{R}$), however, the product of a $P$-space and an rr-good weakly Lindel\"of space is always rr-good. $P$-spaces, $F$-spaces and $U$-spaces play a role, as do Glicksberg's theorem and work by Comfort, Hindman and Negrepontis.

math.GN↗

arXiv2018

The reduced ring order and lower semi-lattices

Every reduced ring $R$ has a natural partial order defined by $a\le b$ if $a^2=ab$; it generalizes the natural order on a boolean ring. The article examines when $R$ is a lower semi-lattice in this order with examples drawn from weakly Baer rings (pp-rings) and rings of continuous functions. Locally connected spaces and basically disconnected spaces give rings $C(X)$ which are such lower semi-lattices. Liftings of countable orthogonal (in this order) sets over surjective ring homomorphisms are studied.

math.RA↗

arXiv1999

$K_1$ of separative exchange rings and C*-algebras with real rank zero

For any (unital) exchange ring $R$ whose finitely generated projective modules satisfy the separative cancellation property ($A\oplus A\cong A\oplus B\cong B\oplus B$ implies $A\cong B$), it is shown that all invertible square matrices over $R$ can be diagonalized by elementary row and column operations. Consequently, the natural homomorphism $GL_1(R) \to K_1(R)$ is surjective. In combination with a result of Huaxin Lin, it follows that for any separative, unital C*-algebra $A$ with real rank zero, the topological $K_1(A)$ is naturally isomorphic to the unitary group $U(A)$ modulo the connected component of the identity. This verifies, in the separative case, a conjecture of Shuang Zhang.

math.RA↗