The Maximal Ring of Quotients of a Triangular Matrix Ring.

On the Maximal Ring of Quotients of C(x)

1. Let QiX) denote the maximal ring of quotients (in the sense of Johnson [4] and Utumi [5]) of the ring C(X) of continuous realvalued functions on the completely regular Hausdorff space X, This ring has been studied by Fine, Gillman, and Lambek [ l] and realized by them as the direct limit of the subrings C(V), Va, dense open subset of X (i.e., the union of these C(F)'s, modulo the obvious equ...

Weak incidence algebra and maximal ring of quotients

Let X, X′ be two locally finite, preordered sets and let R be any indecomposable commutative ring. The incidence algebra I(X,R), in a sense, represents X, because of the wellknown result that if the rings I(X,R) and I(X′,R) are isomorphic, then X and X′ are isomorphic. In this paper, we consider a preordered set X that need not be locally finite but has the property that each of its equivalence...

The Maximal Regular Ideal of a Ring

On the diameter of the commuting graph of the full matrix ring over the real numbers

‎In a recent paper C‎. ‎Miguel proved that the diameter of the commuting graph of the matrix ring $mathrm{M}_n(mathbb{R})$ is equal to $4$ if either $n=3$ or $ngeq5$‎. ‎But the case $n=4$ remained open‎, ‎since the diameter could be $4$ or $5$‎. ‎In this work we close the problem showing that also in this case the diameter is $4$.

When is the ring of real measurable functions a hereditary ring?

‎Let $M(X‎, ‎mathcal{A}‎, ‎mu)$ be the ring of real-valued measurable functions‎ ‎on a measure space $(X‎, ‎mathcal{A}‎, ‎mu)$‎. ‎In this paper‎, ‎we characterize the maximal ideals in the rings of real measurable functions‎ ‎and as a consequence‎, ‎we determine when $M(X‎, ‎mathcal{A}‎, ‎mu)$ is a hereditary ring.